JULIUS WANODflfflEIM '87
Mathematics Dept
COMMENTARY
NEWTON'S PRINCIPIA.
A SUPPLEMENTARY VOLUME.
DESIGNED FOR THE USE OF STUDENTS AT THE UNIVERSITIES.
J. M. F. WRIGHT, A. B.
LATE SCHOLAR OF TRINITY COLLKGK, CAMBRIDGE, AUTHOR OF SOLUTIO.NS
OF THE CAMBRIDGE PROBLEMS, &c. &C.
IN TWO VOLUMES.
VOL. II.
LONDON:
PRINTED FOR T. T. & J. TEGG, 73, CHEAPSIDE;
AND RICHARD GRIFFIN & CO., GLASGOW.
MDCCCXXXIII.
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GLASGOW:
GEORGE BROOKMAX, PHINTER, VILLAFIELU.
INTRODUCTION
VOLUME II.
AND TO THE
MECANIQUE CELESTE.
ANALYTICAL GEOMETRY
1. To determine the position of a point in Jixed space.
Assume any point A in fixed space as known and immoveable, and let
Z' z
three fixed planes of indefinite extent, be taken at right angles to one
another and passing through A. Then shall their intersections A X',
A Y', A Z' pass through A and be at right angles to one another.
ii INTRODUCTION.
This being premised, let P be any point in fixed space ; from P draw
P z parallel to A Z, and from z where it meets the plane- X A Y, draw
z x, z y parallel to A Y, AX respectively. Make
Ax = x, A y = y, P z = z.
Then it is evident that if x, y, z are given, the point P can be found
practically by taking A x = x, A y = y, drawing x z, y z parallel to
AY, AX; lastly, from their intersection, making z P parallel to A Z
and equal to z. Hence x, y, z determine the position of the point P.
The lines x, y, z are called the rectangular coordinates of the point P ;
the point A the origin of coordinates ; the lines A X, A Y, A Z the axes
of coordinates, A X being further designated the axis of x, AY the axis
of y, and A Z the axis of z ; and the planes X A Y, Z A X, Z A Y co
ordinate planes.
These coordinate planes are respectively denoted by
plane (x, y), plane (x, z), plane (y, z) ;
and in like manner, any point whose coordinates are x, y, z is denoted
briefly by
point (x, y, z).
If the coordinates x, y, z when measured along AX, AY, A Z be
always considered positive; when measured in the opposite directions,
viz. along A X' A Y', A Z', they must be taken negatively. Thus ac
cordingly as P is in the spaces
ZAXY, ZAYX', ZAX'Y', ZAY'X;
Z;AXY, Z'AYX', Z'AX'Y', Z'AY'X,
the point P will be denoted by
point (x, y, z), point (— x, y, z), point (— x, — y, z), point (x, — y, z) 5
point (x, y, - z), point (- x, y, - z), point (- x, - y, - z), point (x, - y, - z)
respectively.
2. Given the position of two points (a, ft 7), («', £', /) in Jixed space,
to find the distance let-ween them.
The distance P P' is evidently the diagonal of a rectangular parallele
piped whose three edges are parallel to A X, A Y, A Z and equal to
a s a', (S s j8', 7 s /.
Hence
P P = V (a — a') 2 + (ft — (3') 2 + (7 — /)* • • • • 0)
the distance required.
Hence if P' coincides with A or a', /S', 7' equal zero,
P A = V~^z + /32 + 7* (2)
ANALYTICAL GEOMETRY. iii
3. Calling the distance of any point P (x, y, z) from the origin A of
coordinates the radius-vector, and denoting it by g, suppose it inclined to
the axes AX, AY, A Z or to the planes (y, x}, (x, z), (x, y), by the
angles X, Y, Z.
Then it is easily seen that
x = g cos. X, y = g cos. Y, z = g cos. Z (3)
Hence (see 2)
cos. X = ,—. — 2~r — r~i — »\ > cos- Y = / / z I 2 _i_ — i\ »
so that when the coordinates of a point arc given, the angles which the ra~
dins-vector makes with each of the axes may hence be found.
Again, adding together the squares of equations (3), we have
(x « 4. y 2 + z 2) = g 2 (cos. 2 X + cos. 2 Y + cos. 2 Z).
But
g2 = x2 + y8 + z2 (see 2),
.-. cos. 2 X + cos. 2 Y + cos. 2 Z = 1 . . . . . (5)
which shows that when two of these angles are given the other may be
found.
4. Given two points in space, viz. (a, {3, 7), (of, (Bf, 7'), and one of the
coordinates of any other point (x, y, z) in the straight line that passes
through them, to determine this other point ,• that is, required the equations
to a straight line given in space.
The distances of the point (a, /3, 7) from the points (a', /3', 7'), and
(x, y, z) are respectively, (see 2)
P P = V (a — «')* + (0— /37)1 -f- (7 — /)%
and
PQ= V (a — x) 2 + (0 — y) » + (7 — z) '.
But from similar triangles, we get
(7-z)2: (PQ)2:: (7 - /) 2 : (P F)8
whence
which gives
^a_«')*+(/3_/302H7_z)*=(7-7')U(« — x)2 + (^-y)2}
But since a, a! are independent of /3, $' and vice versa, the two first
terms of the equation,
v INTRODUCTION.
are essentially different from the last. Consequently by (6 vol. 1.)
which give
_/3')2 (7-z)2 = (7-/)2 0-
z — 7 = +
(6)
These results may be otherwise obtained; thus, pgp',is the projection
of the given line on the plane (x, y) &c. as in fig-
P'
Hence
Also
P fl P'
:/ — y::pq:pp'
: : m n : m p'
: : y - /3 : /3'_
z —Y: / — 7 : : p q : p p' : : p r : p m
: : a — X : a — a'.
Hence the general forms of the equations to a straight line given in
space, not considering signs, are
z = «x + b\
' f
z = a y + b'
To find where the straight line meets the planes, (x, y), (x, z), (y, z),
we make
z = 0, y = 0, x = 0,
which give
ANALYTICAL GEOMETRY.
b
z = b'
. b/~ b
a
z = b
b — b'
y : ~V~
which determine the points required.
To find when the straight line is parallel to the planes, (x, y), (x, z),
(y, z), we must make z, y, x, respectively constant, and the equations be
come of the form
2 = C
a'y = ax + b — b'
To find when the straight line is perpendicular to the planes, (x, y),
(x, z) (y, z), or parallel to the axes of z, y, x, we must assume x, y ;
x, z ; y, z; respectively constant, and z, y, x, will be any whatever.
To find the equations to a straight line passing through the origin of
coordinates ; we have, since x = 0, and y = 0, when z = 0,
(9)
z = a yj
5. Tojind the conditions that two straight lines in fixed space may inter-
Sect one another ; and also their point of intersection.
Let their equations be
z = ax + A
z = by + B
z = a' x + A' 1
z = b'y + B'f
from which eliminating x, y, z, we get the equation of condition
a A — a A' _ b' B — b B'
a' — a b' — b
Also when this condition is fulfilled, the point is found from
z = a'A,-"A'. (10)
a — a
6. Tojind the angle /, at which these lines intersect.
Take an isosceles triangle, whose equal sides measured along these
lines equal 1, and let the side opposite the angle required be called i ;
then it is evident that
cos. I = 1 — | i2
as
vi INTRODUCTION.
But if at the extremities of the line i, the points in the intersecting lines
be (x', y', z'), (x", y", z"), then (see 2)
i 2 = (x' — x") 2 + (y — y') 2 + (z' — z") 2
.-. 2 cos. I = 2 — J(x' — x") 2 + (y' — y") * + (z' — z") 2]
But by the equations to the straight lines, we have (5)
z' = a x' -f A ")
z'=by' + BJ
z" — a' x" + A' \
z"=b'y' + B'J
and by the construction, and Art. 2, if (x, y, z) be the point of intersec
tion,
(X_x')*+ (y_y')* + (z — z)2 =
(x— x")2 + (y — y'')2 + (z — z")2 =
Also at the point of intersection,
z = ax+A = a'x + A')
z = by + B = b'y + B'J
From these several equations we easily get
z — z' = a (x — a')
, a M
y — y =-£- (x — x')
z — z" = a' (x — x")
whence by substitution,
/x x/\ 2 i 2 /x x/\ « _i_ — (x x') * =
(X _ x")* + a'2 (X — x") ! + ^ (x— x") 2 =
which give
x — x' =
— /* ft' X
/fl + a/J + ^
W v T b/2
Hence
ANALYTICAL GEOMETRY. vii
Also, since
y — y' = £- (x — x')
arid
z — z' = a (x — x')
z — z" = a' (x — x")
we have
a2 1 . a/2 1 aa'
Hence by adding these squares together we get
2 cos. 1=2-
which gives
1 + aa' +^
cos.I= - - 25 - _ ..... (11;
Ttiis result may be obtained with less trouble by drawing straight lines
from the origin of coordinates, parallel to the intersecting lines ; and then
finding the cosine of the angle formed by these new lines. For the new
angle Is equal to the one sought, and the equations simplify into
z' =ax'=by', z" = a'x" = b' /
z = a x = b y , z=a'x =b'y
i j
From the above general expression for the angle formed by two inter
secting lines, many particular consequences may be deduced.
For instance, required the conditions requisite that two straight lines
given in space may intersect at right angles.
That they intersect at all, this equation must be fulfilled, (see 5)
a' A — a A7 b'B — b B';
a' — a "" b'— b
viii INTRODUCTION.
and that being the case, in order for them to intersect at right angles
t O fD '
we have
T It
1 = — , cos. 1 = 0
and therefore
a a'
7. In the preceding No. the angle between two intersecting lines is
expressed in a function of the rectangular coordinates, which determine
the positions of those lines. " But since the lines themselves would be
given in parallel position, if their inclinations to the planes, (x, y), (x, z),
(y, z), were given, it may be required, from other data, to find the same
angle.
Hence denoting generally the complements of the inclinations of a
straight line to the planes, (x, y), (x, z), (y, z), by Z, Y, X, the problem
may be stated and resolved, as follows :
Required the angle made by the two straight lines, whose angles of 'inclina
tion are Z, Y, X; Z', Y/, X'.
Let two lines be drawn, from the origin of the coordinates, parallel
to given lines. These make the same angles with the coordinate planes,
and with one another, as the given lines. Moreover, making an isosceles
triangle, whose vertex is the origin, and equal sides equal unity, we have
as in (6),
cos. I = 1 — A i2 = 1 — i£(x — x')2 + (y y')2 + (z z')2?
the points in the straight lines equally distant from the origin being
(x, y, z), (x', y', z').
But in this case,
x2 4. y2 -f z2 = 1
x/ 2 + y' 2 + z/ ~ — i
and
x — cos. X, y = cos. Y, z = cos. Z
x' = cos. X', y' = cos. Y', z' = cos. Z'
'. cos. I = x x' + y y' + z z'
= cos. X. cos. X' + cos. Y. cos. Y' + cos. Z. cos. Z'. . (13)
Hence when the lines pass through the origin of coordinates, the same
expression for their mutual inclination holds good ; but at the same time
X, Y, Z ; X', Y', Z', not only mean the complements of the inclinations
to the planes as above described, but also the inclinations of the lines to
the axes of coordinates of x, y, z, respectively.
ANALYTICAL GEOMETRY. ix
8. Given the length (L) of a straight line and the complements of its in
clinations to the planes (x, y), (x, z), (y z), viz. Z, Y, X, tojind the lengths
of its projections upon those planes,
By the figure in (4) it is easily seen that
L projected on the plane (x, y) = L. sin. Z~\
(x, z) = L. sin. Y I . . . (14)
(y, z) = L . sin. X )
9. Instead of determining the parallelism or direction of a straight line
in space by the angles Z, Y, X, it is more concise to do it by means of
Z (for instance) and the angle d which its projection on the plane (x, y)
makes with the axis of x.
For, drawing a line parallel to the given line from the origin of the co
ordinates, the projection of this line is parallel to that of the given line,
and letting fall from any point (x, y, z) of the new line, perpendiculars
upon the plane (x, y) and upon the axes of x and of y, it easily appears,
that
x r= L cos. X = (L sin. Z) cos. 6 (see No. 8)
y = L. cos. Y = (L. sin. Z) sin. 6
which give
cos. X = sin. Z. cos. 6\
cos. Y = sin. Z . sin. 0) ( '
Hence the expression (13) assumes this form,
cos. I = sin. Z . sin. Z' (cos. 6 cos. 6' + sin. 6 sin. 6') + cos. Z cos. Z'
= sin. Z . sin. Z' cos. (6 — 6') + cos. Z . cos. Z' . . . . (16)
which may easily be adapted to logarithmic computation.
The expression (5) is merely verified by the substitution.
10. Given the angle of intersection (I) between two lines in space and
their inclinations to the plane (x, y), tojind the angle at which their pro
jections upon that plane intersect one another.
If, as above, Z, Z' be the complements of the inclinations of the lines
upon the plane, and d, (f the inclinations of the projections to the axis of
x, we have from (16)
cos. (« _ I) = cos. I- cos. Z cos. Z'
sin. Z . sin. Z' v '
This result indicates that I, Z, Z' are sides of a spherical triangle
(radius = 1), 0 — (f being the angle subtended by I. The form may at
once indeed be obtained by taking the origin of coordinates as the center
of the sphere, and radii to pass through the angles of the spherical tri
angle, measured along the axis of z, and along lines parallel to the
given lines.
x INTRODUCTION.
Having considered at some length the mode of determining the posi
tion and properties of points and straight lines in fixed space, we proceed
to treat, in like manner, of planes.
It is evident that the position of a plane is fixed or determinate in posi
tion when three of its points are knowiL Hence is suggested the follow
ing problem.
11. Having given the three points (a, ]3, y), (a', Q'9 /), (a", 0", /') in
space, tojitid the equation to the plane passing through them ; that is, to
Jind the relation between the coordinates of any other point in the plane.
Suppose the plane to make with the planes (z, y), (z, x) the intersec
tions or traces B D, B C respectively, and let P be any point whatever
in the plane ; then through P draw P Q in that plane parallel to B D,
&c. as above. Then
z — QN = PQ' = QQ' cot. D B Z
= y cot. D B Z.
But
QN = AB — NA. cot. C B A
= A B + x cot. C B Z,
.-. z = A B + x cot. C B Z + y cot. D B Z.
Consequently if we put A B = g, and denote by (X, Z), (Y, Z) the
inclinations to A Z of the traces in the planes of (x, z), (y, z) respectively,
we have
z = g + x cot. (X, Z) + y cot. (Y, Z) . , . . (18)
Hence the form of the equation to the plane is generally
ANALYTICAL GEOMETRY. . xi
Now to find these constants there are given the coordinates of three
points of the plane ; that is
7 = A« +B/3 +C
/ = A a! + B /3' + C
7" = A a" + B /3" + C
from which we get
A - 7/3/-//3 + /£"--///3' + 7"/3-7ff/ _ cot /x z) .
A - a /3'-«'/3 + a'^' — a"? + «"/3_«/3" -
R _ r «' — y a + 7' «" — y" a' + y" «•— y «" _ t /Y z^
B - ^F_a'/3 + «'/3"-«"/3' + «"/3 — ^' -
_ /3"(y a' — /«) + g(/a" - /'«') + ^ (/7« - 7 «7/)
a /3/ __ «/ ^ + a' |8" — a" /3' + a" /3 — « /3"
Hence when the trace coincides with the axis of x, we have
C = 0,
and
A = cot. | = 0
/3" (7 a' — / a) + J3 (/ a." — /' a') + F (/' « — 7 a") = 0 >
7 /3' - / /3 + / j9" — /' /3' + 7" /3 _ 7 /3" = 0 j '
1 (/3-/3^) . (/ a"- y" a') + (B' — ff") . (y" a — 7 a^)
' ! - ' ' — ' ' " — '' X " — " '
a
and the equation to the plane becomes
z = By .......... (25)
When the plane is parallel to the plane (x, y),
A = 0, B = 0,
and
z = C ............. (26)
from which, by means of A = 0, B = 0, any two of the quantities 7, 7', y"
being eliminated, the value of C will be somewhat simplified.
Hence also will easily be deduced a number of other particular results
connected with the theory of the plane, the point, and the straight line, of
which the following are some.
To find the projections on the planes (x, y), (x, z), (y, z) of the intersec
tion of the planes,
z=Ax + By + C,
z = A'x + B'y+ C'.
Eliminating z, we have
(A — A')x + (B — B')y + C — C' = 0 .... (27)
which is the equation to the projection on (x, y).
xii INTRODUCTION.
Eliminating x, we get
(A'— A)z + (AB' — A'B)y + AC'— A'C = 0 .... (28)
which is the equation to the projection on the plane (y, z).
And in like manner, we obtain
(B' — B)z + (A'B — AB')x+ BC' — B'C = 0 . . . . (29)
for the projection on the plane (x, z).
To find the conditions requisite that, a plane and straight line shall be
parallel or coincide.
Let the equations to the straight line and plane be
x = a z + A^
y = bz + BJ
z = A' x + B' y + C'.
Then by substitution in the latter, we get
z(A'a+ B'b — 1) + A'A+ B'B + C' = 0.
Now if the straight line and plane have only one point common, we
should thus at once have the coordinates to that point.
Also if the straight line coincide with the plane in the above equation,
z is indeterminate, and (Art. 6. vol. I,)
A'a + B'b — 1 = 0, A'A + B'B + C' = 0 . . . (27)
But finally if the straight line is merely to be parallel to the plane, the
above conditions ought to be fulfilled even when the plane and line are
moved parallelly up to the origin or when A, B, C' are zero. The only
condition in this case is
A' a + B' b = 1 (28)
To Jtnd the conditions that a straight line be perpendicular to a plane
z = Ax+By + C.
Since the straight line is to be perpendicular to the given plane, the
plane which projects it upon (x, y) is at right angles both to the plane
(x, y) and to the given plane. The intersection, therefore, of the plane
(x, y) and the given plane is perpendicular to the projecting plane. Hence
the trace of the given plane upon (x, y) is perpendicular to the projec
tion on (x, y) of the given straight line. But the equations of the traces
of the plane on (x, z), (y, z), are
z= Ax + C, z = By +
or
z = A x -f- L,, z — 15 y -f- ^\
1 C 1 C(
x- Az~ A'y~BZ~ B)
(29)
and if those of the perpendicular be
x = a z + A,\
y = bz + B,J
ANALYTICAL GEOMETRY. xiii
it is easily seen from (11) or at once, that the condition of these traces
being at right angles to the projections, are
A + a = 0, A + b = 0.
To draw a straight line passing through a given point («, /3, 7) at right
angles to a given plane.
The equations to the straight line, are clearly
x _ a + A (z — 7) = 0, y — 0 + B (z — 7) = 0. .' . . (30)
To find the distance of a given point (a, /3, y) from a given plane.
The distance is (30) evidently, when (x, y, z) is the common point in
the plane and perpendicular
But the equation to the plane then also subsists, viz.
from which, and the equations to the perpendicular, we have
z — 7= C — 7 + A a + B/?,
therefore the distance required is
C — 7 + A « + B 0
(31)
A2 + B2
To find the angle I formed by two planes
z = Ax + By+C,
z = A' x + B' y + C'.
If from the origin perpendiculars be let fall upon the planes, the angle
which they make is equal to that of the planes themselves. Hence, if
generally, the equations to a line passing through the origin be
x =r a z )
y = bz/
the conditions that it shall be perpendicular to the first plane are
A + a = 0,
B + b = 0,
and for the other plane
A' + a = 0,
B' + b = 0.
Hence the equations to these perpendiculars are
x + A z = 0
y + Bz = 0
x + A'z = 0
y
'z = 0 \
y z = o, J
xiv INTRODUCTION.
which may also be deduced from the forms (30).
Hence from (11) we get
T _ _ 1 + A A' + B B'
J •= '« " ' ' ' (32>
Hence to find the inclination (s) of a plane with the plane (x, y).
We make the second plane coincident with (x, y), which gives
A' = 0, B' = 0,
and therefore
COS-i= V(1 + A- + B-) ...... (S3)
In like manner may the inclinations (£), (?j) of a plane
z = Ax + By + C
to the planes (x, z), (y, z) be expressed by
COS-^V(l+A* + B*)j ...... (34)
cos< " = V(l + A2 + BV
Hence
cos. 2 s + cos. 2 £ + cos. 2 »j = 1 ...... (35)
Hence also, if E', £', 53' be the inclinations of another plane to (x, y)>
(x, z), (y, z).
COS. I = COS. £ COS. s' + COS. £ COS. <£' + COS. 1) COS. Jj' . . . (36)
Tojind the inclination vofa straight line x = a z + A', y = b z + B',
ft? the plane z = Ax + By+C.
The angle required is that which it makes with its projection upon the
plane. If we let fall from any part of the straight line a perpendicular
upon the plane, the angle of these two lines will be the complement of v.
From the origin, draw any straight line whatever, viz. x = a' z, y = b' z.
Then in order that it may be perpendicular to the plane, we must have
a' = — A, b' = — B.
The angle which this makes with the given line can be found from (11) ;
consequently by that expression
1 — A a — B b ,q7N
sm-"= V(i +a* + b*) v(l + A«+ B'J
Hence we easily find that the angles made by this line and the coor
dinate planes (x, y), (x, z), (y, z), viz. Z, Y, X are found from
^ 1
cos' L - J
ANALYTICAL GEOMETRY. xv
cos. Y = •, • i — t i \ g\ >
cos. X = ^jj a&2 bt> (38)
which agrees with what is done in (3).
TRANSFORMATION OF COORDINATES.
12. To transfer the origin of coordinates to the point (a, ft 7) without
changing their direction.
Let it be premised that instead of supposing the coordinate planes at
right angles to one another, we shall here suppose them to make any
angles whatever with each other. In this case the axes cease to be rec-
*
tangular, but the coordinates x, y, z are still drawn parallel to the axes.
This being understood, assume
x = x' + «, y = / + ft z = z' + 7 (39)
and substitute in the expression involving x, y, z. The result will contain
x', y', z' the coordinates referred to the origin («, ft 7).
When the substitution is made, the signs of a, ft 7 as explained in (1),
must be attended to.
13. To change the direction of the axes from rectangular, without
affecting the origin.
Conceive three new axes A x', A y7, A z', the first axes being supposed
rectangular, and these having any given arbitrary direction whatever.
Take any point ; draw the coordinates x', y', z' of this point, and project
them upon the axis A X. The abscissa x will equal the sum, taken with
their proper signs, of these three projections, (as is easily seen by drawing
the figure) ; but if (x x'), (y, y')» (z> z/) denote the angles between the
axes A x, A x7 ; A y, A y' ; A z, A z' respectively ; these projections
are
x' cos. (x' x), yr cos. (y' x), z7 cos. (z! x).
In like manner we proceed with the other axes, and therefore get
x — x' cos. (x' x) + y' cos. (y' x) + ?! cos. (z' x) *\
y = y' cos. (y' y) + z' cos. (z' y) + x' cos. (x' y) > . . . (40)
z = z' cos. (z' z) -f- y' cos. (y' z) + x' cos. (x' z) )
XVI
INTRODUCTION.
Since (x'x), (x'y), (x'z) are the angles which the staight line A x',
makes with the rectangular axes of x, y, z, we have (5)
cos. 2 (x' x) + cos. s (x' y) + cos. 2 x' z = 1 ^
cos.2 (y'xj 1 + cos.8 (y'y) + cos. 2 (y' z) = 1 V ... (41;
cos. 2 (z' x) + cos.2 (z' y) + cos.2 (z'x) = 1 )
We also have from (13) p.
s.(z'z) i.
S.(Z'Z) )
(42)
I = cos.(x'x)cos.(z'x) -f- cos.(x'y)cos.(z'y ) + cos.(x'
cos.(y'z') ==cos.(y'x)cos.(z'x) + cos.)y'y)cos.(z'y)-f-cos.(y'z)cos.(z'z)
The equations (40) and (41), contain the nine angles which the axes of
x', y', z' make with the axes of x, y, z.
Since the equations (41) determine three of these angles only, six of
them remain arbitrary. Also when the new system is likewise rectangu
lar, or cos. (x'y') = cos. (x'z') = cos. (y' z') = 1, three others of the
arbitraries are determined by equations (42). Hence in that case there
remain but three arbitrary angles.
14. Required to transform the rectangular axe of coordinates to other
rectangular axes, having the same origin, but two of which shall be situated
in a given plane.
Let the given plane be Y A C, of which the trace in the plane (z, x) is
Y'
A C. At the distance A C describe the arcs C Y7, C x, x x' in the planes
C A Y', (z, x), and X' A X. Then if one of the new axes of the coordi
nates be A X', its position and that of the other two, A Y', A Z', will be
determined by C x' = <p , C x = -4/, and the spherical angle x C x7 = 6 =
inclination of the given plane to the plane (z, x).
Hence to transform the axes, it only remains to express the angles
(y/x), (y'x), &c. which enter the equations (40) in terms of 6 *\> and p.
ANALYTICAL GEOMETRY. xvii
By spherics
cos. (x'x) = cos. %}/ cos. 0 + sin. •$> sin. 0 cos. 6.
In like manner forming other spherical triangles, we get
cos. (y' x) = cos. (90° + 0) cos. 4/ + sin. -^ sin. (90° + 0) cos. d
cos. (x' y) = cos. (90° + -^) cos. 0 + sin. (90° + %j/) sin. 0 cos. 6
cos. (y'y) = cos. (90° +^)c
So that we obtain these four equations
cos. (x' x) = cos. 4* cos. 0 +
cos. (y; x) = — sin. -v|/ sin. 0 sn. - cos. cos. / . qv
cos. (x' y) r= — sin. -\|/ cos. 0 +
cos. (y' y) = sin. ^ sin. 0 + cos-
Again by spherics, (since A Z' is perpendicular to A C, and the inclin
ation of the planes Z'A C, (x, y) is 90° — 6) we have
cos (z' x) — sin. -^ sin. & ~\ .
cos. (z 'y) = cos. -^> sin. 6 y ^
And by considering that the angle between the planes Z A C, Z A X', =
90° 4- 6, by spherics, we also get
cos. (x'z) =r — sin. 0 sin. 0 -\
cos. (y'z) = — cos. 0 sin. ^ v (45)
cos. (z'z) = cos. d }
which give the nine coefficients of equations (40).
Equations (41), (42) will also hereby be satisfied when the systems are
rectangular.
15. To find the section of a surface made by a plane.
The last transformation of axes is of great use in determining the na
ture of the section of a surface, made by a plane, or of the section made
by any two surfaces, plane or not, provided the section lies in one plane ;
for having transformed the axes to others, A Z', A X', A Y, the two lat
ter lying in the plane of the section, by the equations (40), and the de
terminations of the last article, by putting z' = 0 in the equation to the
surface, we have that of the section at once. It is better, however, to
make z = 0 in the equations (40), and to seek directly the values of
cos. (x'x), cos. (y'x), &c. The equations (40) thus become
x = x' cos. -4> + y' sm- 4 cos> 6 ~\
y = x' sin. -4/ — y cos. -vj/ cos. 6 V (46)
z = y' sin. 6 )
16. To determine the nature and position of all surfaces of the second
order : or to discuss the general equation of the second order, viz.
ax* + by* + cz 2+ 2dxy + 2exz + 2fyz + gx + hy +iz = k . . (a)
First simplify it by such a transformation of coordinates as shall destroy
xviii INTRODUCTION.
the terms in x y, x z, y z ; the axes from rectangular will become oblique,
by substituting the values (40), and the nine angles which enter these,
being subjected to the conditions (41), there will remain six of them
arbitrary, which we may dispose of in an infinity of ways. Equate to
zero the coefficients of the terms in x' y', x' z', y' z'.
But if it be required that the new axes shall be also rectangular, as this
condition will be expressed by putting each of the equations (42) equal
zero, the six arbitrary angles will be reduced to three, which the three
coefficients to be destroyed will make known, and the problem will thus
be determined.
This investigation will be rendered easier by the following process :
Let x=r a z, y = /3 z be [the equations of the axis of x7 ; then for
brevity making
1 = V (I + a2 + /32)
we find that (3)
cos. (x' x = a 1, cos. (x7 y) = /S 1, cos. x7 z = 1.
Reasoning thus also as to the equations x = a! z, y = $' z of the axis
of y7, and the same for the axis of z', we get
cos. (y'x) = a7!7, cos. (y7y) = /3717, cos. (y7 z) = I7
cos. (z' x) = a77 1", cos. (z7 y) = /3" I77, cos. (z7 z) = I77.
Hence by substitution the equations (40) become
x = 1 a x7 + I7 a' y7 + I77 a7
y = l/3x' +
z = 1 x7 + I7 y7
The nine angles of the problem are replaced by the six unknowns a,
a7, a77, /3, /37, (S/7, provided the equations (41) are thereby also satisfied.
Substitute therefore these values of x, y, z, in the general equation of-
the 2d degree, and equate to zero the coefficients of x' y7, x7 z', y' z7, and
we get
(a a + d 0 + e) a77 + (d a + b (3 + f) $" + e a + f 0 + c = 0 >
(aa77 + d/377 + e) of + (da'7 + b/S77+ f) /37 +e a77 + f/377 + c = 0 J
One of these equations may be found without the others, and by making
the substitution only in part. Moreover from the symmetry of the pro
cess the other two equations may be found from this one. Eliminate a7,
B' from the first of them, and the equations x = a! z, y = /37 z, of the
axis of y7; the resulting equation
(a a + d /3 + e) x + (d a + b /3 + f) y + (e a + f 8 + c] z = 0 . . (b)
is that of a plane (19).
I7 a' y' + I" a" z' -\
I7 & y' + 1" /3" z' V
T y' + l"z'. )
ANALYTICAL GEOMETRY. xix
But the first equation is the condition which destroys the term x'y't
since if we only consider it, a, /?, a', /3', may be any whatever that will
satisfy it ; it suffices therefore that the axis of y' be traced in the plane
above alluded to, in order that the transformed equations may not contain
any term in x' y'.
In the same manner eliminating a", jS", from the 2d equation by means
of the equations of the axis of z', viz. x = a" z, y = /3" z, we shall have
a plane such, that if we take for the axis of z' every straight line which it
will there trace out, the transformed equation will not contain the term in
x' z\ But, from the form of the two first equations, it is evident that this
second plane is the same as the first : therefore, if we there trace the axes
of y' and z' at pleasure, this plane will be that of y' and z', and the
transformed equation will have no terms involving x' y or x z'. The
direction of these axes in the plane being any whatever, we have an in
finity of systems which will serve this purpose; the equation (b) will be
that of a plane parallel to the plane which bisects all the parallels to x,
and which is therefore called the Diametrical Plane.
Again, if we wish to make the term in y' z' disappear, the third equa
tion will give a', @, and there are an infinity of oblique axes which will
answer the three required conditions. But make x', y', z', rectangular.
The axis of x' must be perpendicular to the plane (y z') whose equa
tion we have just found ; and that x = a z, y = /3 z, may be the equa
tions (see equations b) we must have
a« + d/3 + e = (e« + f/3 + c) « . . . . (c)
d a + b 0 + f = (e a + f ,3 + c) /S . . . . (d)
Substituting in (c) the value of a found from (d) we get
{(a — b)fe + (f2 — eVU/33
+ j (a — b) (c — b)e+ (2d2— f2 — e*) e + (2c — a — b)fd} /3J
+ ( (c— a) (c — b) d+ (2e2 — f2 — d2) d + (2b — a — c) f e } 0
+ (a — c) fd + (f2 — d2)e = 0.
This equation of the 3d degree gives for /3 at least one real root ; hence
the equation (d) gives one for a; so that the axis of x' is determined so as
to be perpendicular to the plane (y', z',) and to be free from terms in
x' z', and y'z'. It remains to make in this plane (y, z',) the axes at right
angles and such that the term x' y' may also disappear. But it is evident
that we shall find at the same time a plane (x', z',) such that the axis of y'
is perpendicular to it, and also that the terms in x' y, z' / are not involved.
But it happens that the conditions for making the axis of y' perpendicular
to this plane are both (c) and (d) so that the same equation of the 3d de-
62
xx INTRODUCTION.
gree must give also P. The same holds good for the axis of z. Conse
quently the three roots of the equation J3 are all real, and are the values
of ft /?, |8". Therefore «, a', a", are given by the equation (d). Hence,
There is only one system of rectangular axes 'which eliminates x' y', x' z',
x'y'; and there exists wie in all cases. These axes are called the Princi-
val axes of the Surface.
Let us analyze the case which the cubic in /3 presents.
1. If we make
(a-b)fe + (f2 — e2) d = 0
t.he equation is deprived of its first term. This shows that then one of
the roots of B is infinite, as well as that a derived from equation (d) be
comes e a + f B = 0. The corresponding angles are right angles. One
of the axes, that of z' for instance, falls upon the plane (x, y), and we
obtain its equation by eliminating a and {3 from the equations x = a z,
y = j3 z, which equation is
ex + fy = 0
The directions of y', z' are given by the equation in B reduced to a
quadrature.
Sndly. If besides this first coefficient the second is also = 0, by substi
tuting b, found from the above equation, in the factor of [S 2, it reduces to
the last term of the equation in ft viz.
(a— c) fd + (f2 — ds) e = 0.
These two equations express the condition required. But the coeffi
cient of 8 is deduced from that of B 2 by changing b into c and d into e,
and the same holds for the first and last term of the equation in ft
Therefore the cubic equation is »lso thus satisfied. There exists therefore
an infinity of rectangular systems, which destroy the terms in x' y, x' z',
y' z. Eliminating a and b from equations (c) and (d) by aid of the above
two equations of condition, we find that they are the product of fa — d
and e^ — d by the common factor eda + fd/3 + fe. These factors
are therefore nothing ; and eliminating a and ft we find
fx = dz, ey = d z, e d x + f d y + f e z = 0.
The two first are the equations of one of the axes. The third that oi
a plane which is perpendicular to it, and in which are traced the two
other axes under arbitrary directions. This plane will cut the surface in
a carve wherein all the rectangular axes are principal, which curve is
therefore a circle, the only one of curves of the second order which has
that property. The surface is one then of revolution round the axis
whose equations we have just given.
ANALYTICAL GEOMETRY. xxi
The equation once freed from the three rectangles, becomes of the
form
kz2 + my2-fnx2 + qx + q'y-fq"z = h . . . . (e)
The terms of the first dimension are evidently destroyed by removing
the origin (39). It is clear this can be effected, except in the cas*
where one of the squares x 2, y 2, z 2 is deficient. We shall examine these
cases separately. First, let us discuss the equation
kz2 + my2 + nx2 = h (f)
Every straight line passing through the origin, cuts the surface in two
points at equal distances on both sides, since the equation remains the same
after having changed the signs of x, y, z. The origin being in the middle
of all the chords drawn through this point is a center. The surface therefore
has the property of possessing a center whenever the transformed equation
has the squares of all the variables.
We shall always take n positive : it remains to examine the cases where
k and m are both positive, both negative, or of different signs.
If in the equation (f) k, m, and n, are all positive, h is also positive ;
and if h is nothing, we have x = 0, y =: 0, z = 0, and the surface has
but one point.
But when h is. positive by making x, y, or z, separately equal zero, we
find the equations to an ellipse, curves which result from the -section of
the surface in question by the three coordinate planes. Every plane
parallel to them gives also an ellipse, and it will be easy to show the
same of all plane sections. Hence the surface is termed an Ellip
soid.
The lengths A, B, C, of the three principal axes are obtained by find
ing the sections of the surface through the axes of x, y, and z. Th^e
give
kC2 = h, mB2= h, nA! = h.
from which eliminating k, m and n, and substituting in equation (f) we get
^1-4-^1+ *'- - 1 ")
C* " B 2 "*" A * " I (47)
A'B'z2 + A2 C2y2 + B2C2x2 = Aa B* C2j
which is the equation to an Ellipsoid referred to its center and principal
axes.
We may conceive this surface to be generated by an ellipse, traced in
the plane (x, y), moving parallel to itself, whilst its two axes vary, the
curve sliding along another ellipse, traced in the plane (x, z) as a direct-
6 3
xxii INTRODUCTION.
rix. If two of the quantities A, B, C, are equal, we have an ellipsoid of
revolution. If all three are equal, we have a sphere.
Now suppose k negative, and m and h positive or
k z 2 — my2 — ax2 = — h.
Makings or y equal zero, we perceive that the sections by the planes
(y z) and (x z), are hyperbolas, whose axis of z is the second axis. All
planes passing through the axis of z, give this same curve. Hence the
surface is called an hyperboloid. Sections parallel to the plane (x y) are
always real ellipses, A, B, C V — 1 designating the lengths upon the
axes from the origin, the equation is the same as the above equation ex
cepting the first term becoming negative.
Lastly, when k and h are negative
kz2 + my2 + nx2 = — h;
all the planes which pass through the axis of z cut the surface in hyper
bolas, whose axis of z is the first axis. The plane (x y) does not meet
the surface and its parallels passing through the opposite limits, give
ellipses.' This is a hyperboloid also, but having two vertexes about the
axis of z. , The equation in A, B, C is still the same as above, excepting
that the term in z ' is the only positive one.
When h = 0, we have, in these two cases,
k2* = my2 + nx2 . . . . '. . . • (48)
'the equation to a cone, which cone is the same to these hyperboloids that
an asymptote is to hyperbolas.
It remains to consider the case of k and m being negative. But by a sim
ple inversion in the axes, this is referred to the two preceding ones. The
hyperboloid in this case has one or two vertexes about the axis of x ac
cording as h is negative or positive.
When the equation (e) is deprived of one of the squares, of x l for in
stance, by transferring the origin, we may disengage that equation from
the constant term and from those in y and z ; thus it becomes
kz2 + mys = hx (49)
The sections due to the planes (x z), (x y) are parabolas in the same
or opposite directions according to the signs of k, m, h ; the planes par
allel to these give also parabolas. The planes parallel to that of (y z)
give ellipses or parabolas according to the sign of m. The surface is an
elliptic paraboloid in the one case, and a hyperbolic paraboloid in the
other case. When k = m, it is a paraboloid of revolution.
When h = 0, the equation takes the form
a * z ~ ± b « y 2 = 0
ANALYTICAL GEOMETRY. xxiii
according to the signs of k and m. In the one case we have
z = 0, y = 0
whatever may be the value of x, and the surface reduces to the axis of x,
In the other case.
(a z + b y) (a z — by) = 0
shows that we make another factor equal zero ; thus we have the system
of two planes which increase along the axis of x.
When the equation (e) is deprived of two squares, for instance of x2,
y *, by transferring the origin parallelly to z, we reduce the equation to
kz2 + py + qx = 0 (50)
The sections made by the planes drawn according to the axis of z, are
parabolas. The plane (x y) and its parallels give straight lines par-r
allel to them. The surface is, therefore, a cylinder whose base is a para
bola, or a parabolic cylinder.
If the three squares in (e) are wanting, it will be that of a plane.
It is easy to recognise the case where the proposed equation is decom
posable into rational factors ; the case where it is formed of positive
squares, which resolve into two equations representing the sector of two
planes.
PARTIAL DIFFERENCES.
17. If u = f (x, y, z, &c.) denote any function of the variable x, y, z,
&c. d u be taken on the supposition that y, z, &c. are constant, then is the
result termed the partial difference of u relative to x, and is thus written
/d u\ ,
( j — ) « x-
\d x/
Similarly
rdu,
(
denote the partial differences of u relatively to y, z, &c. respectively.
Now since the total difference of u arises from the increase or decrease
of its variables, it is evident that
xxiv INTRODUCTION.
But, by the general principle laid down in (6) Vol. I, this result may
be demonstrated as follows ; Let
u + du = A + Adx+Bdy + C d z +&c.
A'dx2 + B'dy2+ C'dz' + &c. |
+ Mdx.dy+Ndx.dz+&c.J
Then equating quantities of the same nature, we have
du = Adx+Bdy+Cdz + &c.
Hence when d y, d z, &c. = 0, or when y, z, &c. are considered con
stant
d u = A d x
or according to the above notation
A =
In the same manner it is shown, that
&c.
Hence
« = () d x + (=.) d y + ( ) d z + &c. as before.
Ex. 1. u =r x y z, &c.
du\ /du du
• =z
.•. du = yzdx + xzdy + xydz
du dx dy.dz
or - = --- \- — i + ^ — .
u x y z
Ex. 2. u = x y z, &c. Here as above
in = li . . £y + ii + &c.
u x y z
And in like manner the total difference of any function of any number
of variables may be found, viz. by first taking the partial differences, as in
the rules laid down in the Comments upon the first section of the first
book of the Principia.
But this is not the only use of partial differences. They are frequently
used to abbreviate expressions. Thus, in p. 13, and 14, Vol. II. we
ANALYTICAL GEOMETRY. xxv
learn that the actions of M, /«., p", &c. upon /* resolved parallel to x,
amount to
p'(x'— x) y> (x"—x)
(x'— X)*+(y'_y )>+(£_
•*" (x'"— *) , & MX
"
--[(x"'— x)2 + (/"— y)*-f- (z"'— z)2]* " [(X« + f+ z») 3
retaining the notation there adopted.
But if we make
V(x-x)* + (y'-y)8 + (z — z)2 = e
0, 1
and generally
V(x"-n— X"-m) 2 + (y"-n_y"...m) 2 + ^z''...n_z»...mj 2 Sf
n, m,
and then assume
x = ^'+^ + &c ...... ,.-,?, (A)
0, 1 0,2
+^ + *£ + &c. ... '/ ; ; ,,v
2 1,3
2, 2,4
&C.
we get
._ W' (x'— x) ^ ^' (x"— x)
' ~-
dx
0, 1 0, 2
-
dy
dz
0, 1 0,2
0, 1 0, 2
We also get
^ ^ (x' — x) / d B
0,1
dXx ^"(X"_ x)
, +
1, 2
"'— x ) AtV'(x"— x") /dD
0,3 1,3 2,3
INTRODUCTION.
Hence since (B) has one term less than (A) ; (C) one term less than
) ; and so on ; it is evident that
Cr") ~*.
and therefore that
.
\dx/ \dx/T\dx/ \dx"
See p. 15, Vol. II.
Hence then X is so assumed that the sum of its partial differences re*
lative to x, x', x" &c. shall equal zero, and at the same time abbreviate
the expression for the forces upon p along x from the above complex
formula into
d « (g + x) IfSl^ ,Mx.
dt8 ?\&J~ '~JT*
and the same may be said relatively to the forces resolved parallel to
y, z, &c. &c.
Another consequence of this assumption is
or
For
d x N _ w*(x'— x)y ^"(x"— x)y
— ' s ' T
VK— x')y . ^VV — xQy & __
"~
.
'
3
&C.
Hence it is evident that
t* W— x)(y— y') , /.^'(x"— x)(y— y") &c>
3
^V'(x— x')(y— y") ^>"(x'"-x) (y-yl &c<
§12 ' ?2
^X'(2x'"_x") (y"— y'") M-VCx""— x") (y"— f") + &c>
+ «3 ?*
23
&C.
ANALYTICAL GEOMETRY. xxvii
In like manner it is found that
^'(y'-y)(x-x) + ^"(y"-v)(x-x-Q &c
3 3
y
0, 1 «, 2
/^>';(y"— y') (x— x") i*i*"(y"— y) (*'— x'") „
+ ~p- — + &c.
1, 2 1, 3
&c.
which is also perceptible from the substitution in the above equation of
y for x, x for y ; y' for X', x' for y' ; and so on.
But
(y'-y) (x— x') = (x— x) (y— y')
(y"— y) (x— x") = (x"— x) (y— y")
&c.
consequently
2 x
c
See p. 16. For similar uses of partial differences, see also pp. 22, and
105.
CHANGE OF THE INDEPENDENT VARIABLE.
When an expression is given containing differential coefficients, sucli
as
dj d 2 y
ci x d x
in which the first differential only of x and its powers are to be found, it
shows that the differential had been taken on the supposition that dx is
constant, or that d2x = 0, d ' x = 0, and so on. But it may be re
quired to transform this expression to another in which d*x, d3x shall
appear, and in which d y shall be constant, or from which d 2 y, &c. shall
be excluded. This is performed as follows :
For instance if we have the expression
dy2
1 +
d x1 d y
dx*
the differential coefficients
d y d^y
d'x' dx"
xxviii INTRODUCTION.
may be eliminated by means of the equation of the curve to which we
mean to apply that expression. For instance, from the equation to a
parabola y = a x 2, we derive the values of
dy . d2 y
-j-^- and -T-i
d x dx2
which being substituted in the above formula, these differential coefficients
will disappear. If we consider
dy d* y
dlE' die2
unknown, we must in general have two equations to eliminate them from
one formula, and these equations will be given by twice differentiating the
equation to the curve.
When by algebriacal operations, d x ceases to be placed underneath
d y, as in this form
. _ y(dx' + dy2) (52)
dx* + dy2 — y d«y
the substitution is effected by regarding d x, d y, dzy as unknown; but
then in order to eliminate them, there must be in general the same
number of equations as of unknowns, and consequently it would seem the
elimination cannot be accomplished, because by means of the equation to
the curve, only two of the equations between d x, d y, d 2 y can be ob
tained. It must be remarked, however, that when by means of these two
equations we shall have eliminated d y and d 2 y, there will remain a com
mon factor d x 8, which will also vanish. For example, if the curve is
always a parabola represented by the equation y =. ax , by differentiat
ing twice we obtain
dy = 2axdxOd2y = 2a dx*
and these being substituted in the formula immediately above, we shall
obtain, after suppressing the common factor d x 2,
4 a2 x 2 — Say
The reason why d x2 becomes a common factor is perceptible at once,
for when from a formula which primitively contained
d'y dy
d x 2 ' d x '
we have taken away the denominator of p«J all the terms independent
of ^-2- and V^ must acquire the factor d x 2 ; then the terms which
d x 2 d x
were affected by -r-^» do not contain dx, whilst those affected by j*
ANALYTICAL GEOMETRY. xxix
contain d x. When we afterwards differentiate the equation of the curve,
and obtain results of the form dy = M d x, d 2 y = Ndx2, these values
being substituted in the terms in d2y, and in dy dx, will change them,
as likewise the other terms, into products of d x 2.
What has been said of a formula containing differentials of the two first
orders applying equally to those in which these differentials rise to supe
rior orders, it thence follows that by differentiating the equation of the
curve as often as is necessary, we can always make disappear from the
expression proposed, the differentials therein contained.
The same will also hold good if, beside these differentials which we have
just been considering, the formula contain terms in d £ x, in d 3 x, &c. ;
for suppose that there enter the formula these differentials d x, d y, d " x,
d 2 y and that by twice differentiating the equation represented by y = f x,
we obtain these equations
F (x, y, d y, d x) = 0
F(x,y,dx,dy,d«x,d«y) = 0,
we can only find two of the three differentials d y, d 2 x, d ~ y, and we see
it will be impossible to eliminate all the differentials of the formula ; there
is therefore a condition tacitly expressed by the differential d2x; it is
that the variable x is itself considered a function of a third variable which
does not enter the formula, and which we call the independent variable.
This will become manifest if we observe, that the equation y = f x may
be derived from the system of two equations
x =: F t, y = p t
from which we may eliminate t. Thus the equation
(x — c)s
v — n J —
y • b*
is derived from the system of two equations
x = b t + c, y = a t2,
and we see that x and y must vary by virtue of the variation which t may
undergo. But this hypothesis that x and y vary as t alters, supposes that
there are relations between x and t, and between y and t. One of these
relations is arbitrary, for the equation which we represent generally by
y = f x, for example
a / \ .
y = b - (x — c) *,
if we substitute between x and t, the arbitrary relation,
t3
x ~ ~
xxx
INTRODUCTION.
this value being put in the equation
will change 'it to
y = (x —
an equation which, being combined with this,
ought to reproduce by elimination,
(x — c) «
y = a^—5-r-,
the only condition which we ought to regard in the selection of the varia
ble t.
We may therefore determine the independent variable t at pleasure.
For example, we may take the chord, the arc, the abscissa or ordinate
for this independent variable ; if t represent the arc of the curve, we
have
t = V (dx« + dy2);
if t denote the chord and the origin be at the vertex of the curve, we
have
t = V (x2 + y2);
lastly, if t be the abscissa or ordinate of the curve, we shall have
t = x, or t = y.
The choice of one of the three hypotheses or of any other, becoming in-
dispensible in order that the formula which contains the differentials, may
be delivered from them, if we do not always adopt it, it is even then tacitly
supposed that the independent variable has been determined. For ex
ample, in the usual case where a formula contains only the differentials
d x, d y, d 2 y, d 3 y, &c. the hypothesis is that the independent variable
t has been taken for the abscissa, for then it results that
dx
i = x' 31 =: 1}
d 2-K
5-4 = 0,
d t2
4^1 = 0, &c.
d t -
and we see that the formula does not contain ths second, third, &c. dif
ferentials.
ANALYTICAL GEOMETRY. xxxi
To establish this formula, in all its generality, we must, as above, sup
pose x and y to be functions of a third variable t, and then we have
d y _ d y d x
dT "" cTx* dT'
from which we get
ai = ini • • • (53)
ft
taking the second differential of y and operating upon the second meinbei
as if a fraction, we shall get
d x d * y d y d8 x
d8 y _ d~t ' cU dT' d t
dx " " dx2
d t2
In this expression, d t has two uses; the one is to indicate that it is
the independent variable, and the other to enter as a sign of algebra.
In the second relation only will it be considered, if we keep in view that
t is the independent variable. Then supposing d t 2 the common factor,
the above expression simplifies into
d2y _ dxd2y — dy d2x
dx = d x8 '
and dividing by d x, it will become
d* y __ d x d* y — dyd2x
die2 = dx3
Operating in the same way upon the equation (53), we see that in
taking t as the independent variable, the second member of the equation
ought to become identical with the first ; consequently we have only one
change to make in the formula which contains the differential coefficients
d y d 2 y d 2 v
-j » ~T~i > V1Z- to replace Jt by
dxdzy — dyd2x
d x 2 * V /
To apply these considerations to the radius of curvature which is given
by the equation See p. 61. vol. I.)
i
dx
xxxii INTRODUCTION.
if we wish to have the value of R, in the case where t shall be the inde
pendent variable, we must change the equation to
n (!*&)' . ;;'•;.
dx d2 y — d y d - x *
dx3
and observing that the numerator amounts to
(dx' + dy')*
dx3
we shall have
dy'f
R -
cTx^d'y — dy'd'x '
This value of R supposes therefore that x and y are functions of a third
independent variable. But if x be considered this variable, that is to say,
if t = x, we shall have d 2 x =0, and the expression again reverts to the
common one
(dx'+dy*)* V1 + dx')
dxd'y d2y
dx'
But if, instead of x for the independent variable, we wish to have the
ordinate y, this condition is expressed by y = t ; and differentiating this
equation twice, we have
The first of these equations merely indicates that y is the independent
variable, which effects no change in the formula ; but the second shows
us that d * y ought to be zero, and then the equation (55) becomes
_(dx« + dy«)* (56)
dy d2x
We next remark, that when x is the independent variable, and
consequently d2x = 0, this equation indicates that d x is constant.
Whence it follows, that generally the independent variable has always
a constant differential.
Lastly, if we take the arc for the independent variable, we shall have
dt = V (dx2 + dy1);
Hence, we easily deduce
dx* , d_y2 ..
+ » '
ANALYTICAL GEOMETRY. xxxiii
differentiating this equation, we shall regard d t as constant, since t is the
independent variable ; we get
2 d xd'x 2dy d2y
~d~F~ dt» "
which gives
dxd2x = — d y d 2 y •
Consequently, if we substitute the value of d * x, or that of d z y, in the
equation (55), we shall have in the first case
' ' . V(dx* + dy )
II A — 1—5 il A . • I <1 I I
'
(d x + d y 2) d " y d * y
and in the second case,
j
„ (dx2-f dy'2)2 , V (dx 2 + dy 2) , .„,...
li = — 7-5 — ,J ' , j d y = > n J—L d y . (58)
(d x + d y ) d x J d x
In what precedes, we have only considered the two differential coeffi
cients
(Ii ily .
but if the formula contain coefficients of a higher order, we must, by
means analogous to those here used, determine the values of
^Xf^ &c
do vl I jj CVV.«
x J d x *
which will belong to the case where x and y are functions of a third in
dependent variable.
PROPERTIES OF HOMOGENEOUS FUNCTIONS.
IfMdx + Ndy -f Pdt + &o. = d z, be a homogeneous function of
any number of variables, x, y, t, &c. in which the dimension of each term is
n, then is
MX + Ny + Pt + &c. = nz.
For let M d x + N d y be the differential of a homogeneous function
z between two variables x and y ; if we represent by n the sum of the
exponents of the variables, in one of the terms which compose this func
tion, we shall have therefore the equation
Mdx + Ndy = dz.
y
Making * = q, we shall find (vol. I.)
F(q) X x" = z;
xxxiv INTRODUCTION.
and replacing, in the above equation, y by its value q x, and calling M'
N', what M and N then become, that equation transforms to
M' d x + N' d. q x = d z ;
and substituting the value of z, we shall have
M' d x -f N' d (q z) = d (x " F. q.)
But d (q z) =: q d x + x d q. Therefore
(M' + N'q) dx + N'xdq = d (xn F. q).
But, (M' + N' q) d x being the differential of x " F q relatively to x, we
have (Art. 6. vol. 1.)
M' + N'q = nx"-1 X F. q.
If in this equation y be put for q x, it will become
M + N-£ = nx'-'F. q,
x
or
Mx+Ny = nz.
This theorem is applicable to homogeneous functions of any number of
variables ; for if we have, for example, the equation
M d x + Ndy+ Pdtrrdz,
in which the dimension is n in every term, it will suffice to make
y t
— = q> — = r
x ^ x
to prove, by reasoning analogous to the above, that we get z — x" F (q, r),
and, consequently, that
Mx + Ny+Pt = nz (59)
and so on for more variables.
THEORY OF ARBITRARY CONSTANTS.
An equation V = 0 between x, y, and constants, may be considered as
the complete integral of a certain differential equation, of which the order
depends on the number of constants contained in V = 0. These constants
are named arbitrary constants, because if one of them is represented by a,
and V or one of its differentials is put under the form f (x, y) = a, we see
that a will be nothing else than the arbitrary constant given by the integra
tion of d f (x, y). Hence, if the differential equation in question is of the
order n, each integration introducing an arbitrary constant, we have
V == 0, which is given by the last of three integrations, and contains, at
ANALYTICAL GEOMETRY. xxxv
least, n arbitrary constants more than the given differential equation. Let
therefore
F(x,y) = 0,F x,y, = 0,F x,y,, = 0 &c. (a)
be the primitive equation of a differential equation of the second order
and its immediate differentials.
Hence we may eliminate from the two first of these three equations,
the constants a and b, and obtain j
If, without knowing F (x, y) = 0, we find these equations, it will be
sufficient to eliminate from them -r* , to obtain F (x, y) = 0, which will
dx
be the complete integral, since it will contain the arbitrary constants a, b.
If, on the contrary, we eliminate these two constants between the
above three equations, we shall arrive at an equation which, containing
the same differential coefficients, may be denoted by
d d 2
But each of the equations (b) will give the same. In fact, by eliminating
the constant contained in one of these equations and its immediate differ
ential, we shall obtain separately two equations of the second order,
which do not differ from equation (c) otherwise than the values of x and
of y are not the same in both. Hence it follows, that a differential equa
tion of the second order may result from two equations of the first order
which are necessarily different, since the arbitrary constant of the one is
different from that of the other. The equations (b) are what we call the
first integrals of the equation (c), which is independent, and the equation
F (x, y) = 0 is the second integral of it.
Take, for example, the equation y = a x + b, which, because of its
two constants, rnay be regarded as the primitive equation of an equation
of the second order. Hence, by differentiation, and then by elimination
of a, we get
d y d y . i
-5-^- = a , y = x -r— -f b.
dx dx
These two first integrals of the equation of the second order which we
are seeking, being differentiated each in particular, conduct equally, by
1 2
the elimination of a, b, to the independent equation -. — ^ = 0. In the
c3
xxxvi INTRODUCTION.
case where the number of constants exceeds that of the required arbitrary
constants, the surplus constants, being connected with the same equations,
do not acquire any new relation. Required, for instance, the equation of
the second order, whose primitive is
differentiating we get
iZ = ax + b.
dx
The elimination of a, and then that of b, from these equations, give
separately these two first integrals
5-Z = ax + b, y = XC|^ — \ ax2 + c . . . (d)
dx ' J dx
Combining them each with their immediate differentials, we arrive,
d 2 y
by two different ways, at -, — -„ = a. If, on the contrary, we had elimi
nated the third constant a between the primitive equation and its imme
diate differential, that would not have produced a different result; for
we should have arrived at the same result as that which would lead to
the elimination of a from the equations (d), and we should then have
1 2 *J
fallen upon the equation x -j-fz = -^ — b, an equation which reduces
d2y
to -j — ~ = a by combining it with the first of the equations (d).
Let us apply these considerations to a differential equation of the third
order : differentiating three times successively the equation F (x, y) = 0,
we shall have
F (x,y, to = 0, F(x,y, d^, ^) = 0, F (x, y,-^,^ , ^ = 0
V >t7'dx/ V ' Jy dx' dxV \ J dx dx* dx3/
These equations admitting the same values for each of the arbitrary
constants contained by F (x, y) z= 0, we may generally eliminate these
constants between this latter equation and the three preceding ones, and
obtain a result which we shall denote by
c I dy d2y d3y\
f(x'y>si'd/"d^) =° ...... w
This will be the differential equation of the third order of F (x, y) = 0.
and whose three arbitrary constants are eliminated ; reciprocally,
F (x, y) =r 0, will be the third integral of the equation (e).
If we eliminate successively each of the arbitrary constants from the
ANALYTICAL GEOMETRY. xxxvii
equation F (x, y) = 0, and that which we have derived by differentiation,
we shall obtain three equations of the first order, which will be the "second
integrals of the equation (e).
Finally, if we eliminate two of the three arbitrary constants by means
of the equation F (x, y) = 0, and the equations which we deduce by two
successive differentiations, that is to say, if we eliminate these constants
from the equations
F (x,y) = 0, F (*,y, £) = 0, F (x,y, 1|, £) = 0 . . (f)
we shall get, successively, in the equation which arises from the elimina
tion, one of the three arbitrary constants ; consequently, we shall have as
many equations as arbitrary constants. Let a, b, c, be these arbitrary
constants. Then the equations in question, considered only with regard
to the arbitrary constants which they contain, may be represented by
p c = 0, <p b = 0, <f> a = 0 (g)
Since the equations (f) all aid in the elimination which gives us one of
these last equations, it thence follows that the equations (g) will each be
of the second order; we call them the first integrals of the equation (e).
Generally, a differential equation of an order n will have a number n
of first integrals, which will contain therefore the differential coefficients
from -T* to , D_/t inclusively; that is to say, a number n_1 of differential
( 1 X. (I X
coefficients ; and we see that then, when these equations are all known,
to obtain the primitive equation it will suffice to eliminate from these equa
tions the several differential coefficients.
PARTICULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS.
It is easily seen that a particular integral may always be deduced from
the complete integral, by giving a suitable value to the arbitrary con
stant.
For example, if we have given the equation
xdx + ydy = dyVx* + y2 — a*,
whose complete integral is
y + c = V (x2 + y2 — a'),
whence (for convenience, by rationalizing,) we get
c2
xxxviii INTRODUCTION.
and the complete integral becomes
2 cy + c2 — x2 + a2 = 0 . . . . (i)
Hence, in taking for c an arbitrary constant value c = 2 a, we shall
obtain this particular integral
2 cy + 5 a2 — x2 = 0,
which will have the property of satisfying the proposed equation (h) as
well also as the complete integral. In fact, we shall derive from this
particular integral
x2 — 5 ag cl y x_
~2~c ' d x =: "c '
these values reduce the proposed to
an equation which becomes identical, by substituting in the second mem
ber, the value of c2, which gives the relation c = 2 a. Let
Mdx + Ndy = 0,
be a differential equation of the first order of a function of two variables
x and y ; we may conceive this equation as derived by the elimination of
a constant c from a certain equation of the same order, which we shall
represent by
m d x + n d y = 0,
and the complete integral
F (x, y, c) = 0,
which we shall designate by u. But, since every thing is reduced to
taking the constant c such, that the equation
Mdx + Ndy = 0,
may be the result of elimination, we perceive that is at the same time
permitted to vary the constant c, provided the equation
Mdx + Ndy = 0,
holds good ; in this case, the complete integral
F (x, y, c) = 0
will take a greater generality, and will represent an infinity of curves of
the same kind, differing from one another by a parameter, that is, by a
constant.
Suppose therefore that the complete integral being differentiated, by
considering c as the variable, we have obtained
<' y = (ai) " * + (il) de
ANALYTICAL GEOMETRY. xxxix
an equation which, for brevity, we shall write
d y = p d x + q d c (k)
Hence it is clear, that if p remaining finite, q d c is nothing, the result
of the elimination of c as a variable from
F (x, y, c) = 0,
and the equation (k), will be the same as that arising from c considered
constant, from
F (x, y, c) = 0,
and the equation
d y = p d x
(this result is on the hypothesis
Mdx+Ndy = 0),
for the equation (k), since
q d c = 0,
does not differ from
dy = p d x;
but in order to have
q d c = 0,
one of the factors of this equation -sz constant, that is to say, that we
have
d c = 0, or q =. 0.
In the first case, d c =r 0, gives c = constant, since that takes place
for particular integrals ; the second case, only therefore conducts to a par
ticular solution. But, q being the coefficient of d c of the equation (k),
we see that q = 0, gives
dx "
This equation will contain c or be independent of it. If it contain c,
there will be two cases ; either the equation q = 0, will contain only c
and constants, or this equation will contain c with variables. In the first
case, the equation q = 0, will still give c = constant, and in the second case,
it will give c = f (x, y) ; this value being substituted in the equation
F (x, y, c) = 0, will change it into another function of x, y, which will
satisfy the proposed, without being comprised in its complete integral,
and consequently will be a singular solution ; but we shall have a parti
cular integral if the equation c = f (x, y), by means of the complete "n-
tegral, is reduced to a constant.
c4
xl INTRODUCTION.
When the factor q = 0 from the equation q d c = 0 not containing
the arbitrary constant c, we shall perceive whether the equation q =: 0
gives rise to a particular solution, by combining this equation with the
complete integral. For example, if from q = 0, we get x = M, and put
this value in the complete integral F (x, y, c) = 0, we shall obtain
c = constant = B or c = fy;
In the first case, q = 0, gives a particular integral j for by changing c
into B in the complete integral, we only give a particular value to the
constant, which is the same as when we pass from the complete integral
to a particular integral. In the second case, on the contrary, the value
f y introduced instead of c in the complete integral, will establish between
x and y a relation different from that which was found by merely re
placing c by an arbitrary constant. In this case, therefore, we shall have
a particular solution. What has been said of y, applies equally to x.
It happens sometimes that the value of c presents itself under the form
— : this indicates a factor common to the equations u and U which is ex
traneous to them, and which must be made to disappear.
Let us apply this theory to the research of particular solutions, when
the complete integral is given.
Let the equation be
y dx — xdy = a V^dx2 -f- dy!)
of which the complete integral is thus found.
Dividing the equation by d x, and making
we obtain
y — px = a V(l + p*).
Then differentiating relatively to x and to p, we get
, . a p d p
dy — pdx — xdp = V(1f+pt) ;
observing that
dy ='pdx,
this equation reduces to
, a p d p A
p-
and this is satisfied by making d p = 0. This hypothesis gives p = con
stant s= c, a value which being put in the above equation gives
ANALYTICAL GEOMETRY. xli
y — ex = a V(\ + c') (!)
This equation containing an arbitrary constant c, which is not to be
found in the proposed equation, is the complete integral of it.
This being accomplished, the part q d c of the equation d y = p d x +
q d c will be obtained by differentiating the last equation relatively to c
regarded as the only variable. Operating thus we shall have
, a c d c
xdc+ =0;
consequently the coefficients of d c, equated to zero, will give us
ac
x =
he2)
To find the value of c, we have
— a « r 2
— il V |
which gives
and
- V(a2 — x2)'
by means of this last equation, eliminating the radical of the equation (m)
we shall thus obtain
c =
This value and that of V (I + c2) being substituted in the equation (D
will give us
x2 a2
V(a2 — x2) = : V(a2 — xz)
whence is derived
y = V(a! — x2),
an equation which, being squared, will give us
y2 = a* — x2;
and we see that this equation is a particular solution, for by differentiating
it we obtain
x d x
d y = -- ;
y
this value and that of V(x2 + y=), being substituted in the equation
originally proposed, reduce it to .
a2 = a8.
In the application which we have just given, we have determined the
xbi INTRODUCTION.
/d y\
value of c by equating to zero the differential coefficient (-r^J. This
process may sometimes prove insufficient. In fact, the equation
being put under this form
Adx + Bdy + Cdc = 0
where A, B, C, are functions of x and y, we shall thence obtain
d - _— dx - dc (o)
B 1 C J
d x = — -jr-d y — ir"c (P)
and we perceive that if all that has been said of y considered a function of
x, is applied to x considered a function of y, the value of the coefficient of
d c will not be the same, and that it will suffice merely that any factor of B
destroys in C another factor than that which may destroy a factor of A,
in order that the value of the coefficient of d c, on both hypotheses, may
appear entirely different. Thus although very often the equations
give for c the same value, that will not always happen ; the reason of
which is, that when we shall have determined c by means of the equation
^=0,
dc
d x
it will not be useless to see whether the hypothesis of -= — gives the same
result.
Clairaut was the first to remark a general class of equations susceptible
of a particular solution ; these equations are contained in the form
dy .p, dy
y = -f^-x + F. -j^-
d x f d x
an equation which we shall represent by
y = px + Fp ......... (r)
By differentiating it, we shall find
tins equation, since d y = p d x, becomes
ANALYTICAL GEOMETRY. xliii
and since d p is a common factor, it may be thus written :
We satisfy this equation by making d p = 0, which gives p = const.
= c; consequently, by substituting this value in the equation (r) we
shall find
y = ex + F c .
This equation is the complete integral of the equation proposed, since
an arbitrary constant c has been introduced by integration. If we differ
entiate relatively to c we shall get
Fcx) ,
- tlc-
Consequently, by equating to zero the coefficients of d c, we have
d Fc
* + ^ = °>
which being substituted in the complete integral, will give the particular
solution.
THE INTEGRATION OF EQUATIONS OF PARTIAL DIFFERENCES.
An equation which subsists between the differential coefficients, com
bined with variables and constants, is, in general, a partial differential
equation, or an equation of partial differences. These equations are thus
named, because the notation of the differential coefficients which they
contain indicates that the differentiation can only be effected partially ;
that is to say, by regarding certain variables as constant. This supposes,
therefore, that the function proposed contains only one variable.
The first equation which we shall integrate is this ; viz.
/d z\
(dx) =a'
If contrary to the hypothesis, z instead of being a function of two vari
ables x, y, contains only x, we shall have an ordinary differential equation,
which, being integrated, will give
z = a x + c
but, in the present case, z being a function of x and of y, the ys con
tained in z have been made to disappear by differentiation, since differen-
xhv INTRODUCTION.
tiating relatively to x, we have considered y as constant. We ought,
therefore, when integrating, to preserve the same hypothesis, and suppose
that the arbitrary constant is m general a function of y ; consequently, we
shall have for the integral of the proposed equation
z = ax + py.
Required to integrate the equation
.
in which X is any function of x. Multiplying by d x, and integrating,
we get
z =/Xdx + py.
For example, if the function X were x2 + a2, the integral would be
z = ^ + a2x + ?y.
In like manner, it is found that the integral of
is
z = x Y + <p y .
Similarly, we shall integrate every equation in which (:rr) is equal to
a function of two variables x, y. If, for example,
/d zx _ x
Vd x/ ' " V a y + x 2 '
considering y as constant, we integrate by the ordinary rules, making the
arbitrary constant a function of y. This gives
z = V (ay + x2) + py.
Finally, if we wish to integrate the equation
' ' V(y2 — x2)
regarding y as constant, we get
i x ,
z = sm.~ l --- f- 9 y •
*/
Generally to integrate the equation
we shall take the integral relatively to x, and adding to it an arbitrary
function of y, as the constant, to complete it, we shall find
z = /T(x, y) dx + ty.
ANALYTICAL GEOMETRY. xlv
Now let us consider the equations of partial differences which contain
two differential coefficients of the first order ; and let the equation be
in which M and N represent given functions of x, y. Hence
M
substituting this value in the formula
dz =
•o
fd
which has no other meaning than to express the condition that z is a
function of x and of y, we obtain
/d z\ ( , M
tlz = (dx) |dx- -N
or
/d z\ Ndx — Mdy
d z = ( -j— ) - -
Vdx/
N
Let X be the factor proper to make Ndx — Mdya complete differ
ential d s ; we shall have
X (N d x — M d v) = d s.
By means of this equation, we shall eliminate Ndx — Mdy from the
preceding equation, and we shall obtain
, 1 /d z\ ,
d z = -XT. (T— ). d s.
X N \dx/
Finally, if we remark that the value of (Cj— ) is indeterminate, we may
take it such that ^-^ . ( ^ — \ d s may be integrable, which would make it
a function of s ; for we know that the differential of every given function
of s must be of the form F s . d s. It therefore follows, that we may
assume
z
an equation which will change the preceding one into
d z = F s . d s
which gives
z = 9 s.
xlvi INTRODUCTION.
Integrating by this method the equation
/d z\ /d z\
X [T— 1 — V (-] — ) = 0
\dy/ . J VI x/
we have in this case
M = - y,
and
N = x;
consequently
d s = X (x d x + y d y).
It is evident that the factor necessary to make this integrable is z.
Substituting this for X and integrating, we get
s = x2 + yz.
Hence the integral of the proposed equation is
z = f (x= + y2).
Now let us consider the equation
.
in which P, Q, R are functions of the variables x, y, z ; dividing it by P
and making
Q -M -5-N
p _ ivi, p .
we shall put it under this form :
+N = 0;
y
and again making
and
it becomes
p + M q + N = 0 ........... (a)
This equation establishes a relation between the coefficients p and q of
the general formula
d
= pdx + qdy;
without which relation p and q would be perfectly arbitrary, for as it has
been already observed, this formula has no other meaning than to indicate
that z is a function of two variables x, y, and that function may be any
ANALYTICAL GEOMETRY. xlvii
whatever ; so that we ought to regard p and q as indeterminate m ihis last
equation. Eliminating p from it, we shall obtain
dz + Ndx = q(dy-—Mdx)
and q will remain always indeterminate. Hence the two members of this
equation are heterogeneous (See Art. 6. vol. 1), and consequently
dz + Ndx = 0, dy — M d x = 0 ..... (b)
If P, Q, R do not contain the variable z, it will be the same of M and
N ; so that the second of these equations will be an equation of two varia
bles x and y, and may become a complete differential by means of a factor
A. This gives
X (d y — M d x) = 0.
The integral of this equation will be a function of x and of y, to wluca
we must add an arbitrary constant s ; so that we shall have
F (x> y) = s;
whence we derive
y = f (x, s).
Such will be the value of y given us by the second of the above equa
tions; and to show that they subsist simultaneously we must substitute
this value in the first of them. But although the variable y is not shown,
it is contained in N. This substitution of the value of y just found,
amounts to considering y in the first equation as a function of x and of
the arbitrary constant s. Integrating therefore this first equation on that
hypothesis we find
z = — yN d x + <p s.
To give an example of this integration, take the equation
and comparing it with the general equation (a), we have
M = 2- , N = — — V (x2 + y2).
x x J
These values being substituted in the equations (b) will change them to
d z — — V (x* + y2) d x = 0, d y — 2- d x = 0
X X
Let A be the factor necessary to make the last of these integrable, and
we have
x(dy-I-dx) = 0,
or rather
xlviii INTRODUCTION.
1 V
which is integrable when X = — ; for then the integral is -J = constant.
Put therefore
Z=s
X
and consequently
y = s x.
By means of this value of y, we change the first of the equation!*
(c) into
or rather into
* x — sx ,
d z — a - . d x = 0,
Integrating on the supposition that s is constant, we. shall obtain
z = a/dx V (1 + s2) + <p s
and consequently
z = a x V (1 + s2) + <p s.
Substituting for s its value we get
= a
In the more general case where the coefficients P, Q, R of the equation
contain the three variables x, y, z it may happen that the equations
(.b) contain only the variables which are visible, and which consequently
we may put under the forms
d z = f (x, z) d x = 0, d y = F (x, y) d x.
These equations may be treated distinctly, by writing them as above,
z =/f(x,z)dx + z, y =/F (x,y) dx + <Dy
for then we see we may make z constant in the first equation and y in
the second ; contradictory hypotheses, since one of three coordinates
x, y, z cannot be supposed constant in the first equation without its being
not constant in the second.
Let us now see in what way the equations (b) may be integrated in the
case where they only contain the variables which are seen in them.
Let p and X be the factors which make the equations (b) integrable.
If their integrals thus obtained be denoted by U and by V, we have
A (d z + N d x) = d U, A (d y — M d x) - d V.
ANALYTICAL GEOMETRY. xlix
By means of these values the above equation will become
dU = q-dV . ... .-. . . (d).
Since the first member of this equation is a complete differential the
second is also a complete differential, which requires q — to be a function
of V. Represent this function by <f> V. Then the equation (d) will
become
dU = pV.dV
which gives, by integrating,
U = <i>V.
Take, for example, the equation
/d z\ . (^-\ —
Xjr \dx/ \dy/ "" '
which being written thus, viz.
X/f\ "7» rr
fV ^\ « n
-i — I =r U
we compare it with the equation
and obtain
M = X-, N = — -
y x
By means of these values the equations (b) becomes
dz— -. dx = 0,dy — ~dx = 0;
x y
which reduce to
xdz — zdx = 0,ydy — xdx = 0.
The factors necessary to make these integrable are evidently — ^ and 2.
JH
«f
Substituting which and integrating, we find— and y 2 — x2 for the in-
X
tegrals. Putting, therefore, these values for U and V in the equation
U = * V, we shall obtain, for the integral of the proposed equation,
- = cD (y 2 _ x 2)
X
It must be remarked, that, if we had eliminated q instead of p, the equa
tions (b) would have been replaced by these
Mdz + Ndy=±0,dy — Mdx = 0. . . . (e)
and since all that has been said of equations (b) applies equally to these,
d
I INTRODUCTION.
it follows that, in the case where the first of equations (b) was not in-
tegrable, we may replace those equations by the system of equations (e),
which amounts to employing the first of the equations (e) instead of the
first of the equations (b).
For instance, if we had
/d z\ /d z\
this equation being divided by a z and compared with
will give us
a ' a z
and the equations (b) will become
XV X
d z H *•- dx = 0,dy H dx = 0;
r a z a
which reduce to
azdz + xydx=rO,ady + xdx = 0 . • • (0
The first of these equations, which, containing three variables, is not
immediately integrable, we replace by the first of the equations (e), and
we shall have, instead of the equations (f), these
d z + — — d y = 0,ady + xdx = 0;
a a z J
which reduce to
2ydy — 2zdz = 0,2ady + 2xdx = 0;
equations, whose integrals are
y2 — z8, 2ay + x2-
These values being substituted for U and V, will give us
y2 — z1 = <p (2ay + x2).
It may be remarked, that the first of equations (e) is nothing else than
the result of the elimination of d x from the equations (b) .
Generally we may eliminate every variable contained in the coefficients
M, N, and in a word, combine these equations after any manner what
ever ; if after having performed these operations, and we obtain two in
tegrals, represented by U = a, V = b, a and b being arbitrary constants,
we can always conclude that the integral is U = * V. In fact, since
a and b are two arbitrary constants, having laken b at pleasure, we may
compose a in terms of b in any way whatsoever ; which is tantamount to
saying that we may take for a an arbitrary function of b. This condition
will be expressed by the equations a = <p (b). Consequently, we shall
ANALYTICAL GEOMETRY. 11
have the equations U = <p b, V = b, in which x, y, z represent the same
coordinates. If we eliminate (b) from these equations, we shall obtain
U = pV.
This equation also shows us that in making V = b, we ought to have
U = f b = constant ; that is to say, that U and V are at the sanie time
constant; without which a and b would depend upon one another, where
as the function p is arbitrary. But this is precisely the condition expressed
by the equations U = a, V = b.
To give an application of this theorem, let ' ^
d z\ /d
Dividing by z x and comparing it with the general equation we
have
M = — £, N = — ?—;
ZX
and the equations (b) give us
dz — — dx = 0,dy-fdx = 0
zx J r x
or
zxdz — y * d x = 0, xdy + ydz=0.
The first of these equations containing three variables we shall not at
tempt its integration in that state ; but if we substitute in it for y d x its
value derived from the second equation, it will acquire a common factor
x, which being suppressed, the equation becomes
z d z + y d y • = 0,
and we perceive that by multiplying by 2 it becomes integrable. rl he
other equation is already integrable, and by integrating we find
z 2 + y * = a, xy=b,
whence we conclude that
z2 + y' = Pxy.
We shall conclude what we have to say upon equations of partial differ
ences of the first order, by the solution of this problem.
Given an equation which contains an arbitrary function of one or more
variables, tojind the equation of partial differences "which produced it.
Suppose we have
z= F(x* + y2).
Make
x2 + y2 = u .......... (0
and the equation becomes .
z = Fu.
49
Hi INTRODUCTION.
The differential of F u must be of the form f u . d u. Conse
quently
d z = d u. <p u
If we take the differential of z relatively to x only, that is to say, in
regarding y as constant, we ought to take also d u on the same
hypothesis. Consequently, dividing the preceding equation by d x,
we get
'd z\ /d U'
Again, considering x as constant and y as variable, we shall similarly
find
(—} = /d— \ 9 u
Vdy/ \dy/ '
But the values of these coefficients are found from the equation (f)f
which gives
/d u\ /d u\
I j — J = 2 x , ( -j — ) = 2 y .
\d x/ ' \d y/ *
Hence our equations become
fd z>
(dz\ _ /dz\ rt
dx) =2x?u,(a7)=2y?u;
and eliminating <p u from these, we get the equation required ; viz.
d z\ /d
= x
As another example, take this equation
z8 + 2 ax = F (x y).
Making
x — y = u ,
It becomes
zs + 2ax=Fu
and differ ntiating, we get
d(zs-f-2ax) = du?u.
Then taking the differential relatively to x, we have
and similarly, with regard to y, we get
0 /d z\ /d u
82
ANALYTICAL GEOMETRY. "liii
But since
x — y = u
u
. .
'
which, being substituted in the above equation, gives us
and eliminating <p u from these, we have the equation required ; viz.
We now come to
EQUATIONS OF PARTIAL DIFFERENCES OF THE SECOND ORDER.
Aii equation of Partial Differences of the second order in which z is a
function of two variables x, y ought always to contain one or more of the
differential coefficients
independently of the differential coefficients which enter equations of the
first order.
We shall merely integrate the simplest equations of this kind, and shall
begin with this, viz.
Multiplying by d x and integrating relatively to x we add to the inte
gral an arbitrary function of y ; and we shall thus get
/dz\
(die)
Again multiplying by d x and integrating, the integral will be com
pleted when we add another arbitrary function of y, viz. -^ y. We thus
obtain
z = x p y -f ^ y.
Now let us integrate the equation.
— • P
d3
liv INTRODUCTION.
in which P is any function of x, y. Operating as before we first obtain
(d z\
diJ =/Pdx + ?y;
and the second integration gives us
z = /£/Pdx -f- 9y] dx
In the same manner we integrate
P
- *
^dy
and find
The equation
II
must be integrated first relatively to one of the variables, and then rela
tively to the other, which will give
y + /Pdx}dy .
In general, similarly may be treated the several equations
_ p
in which P, Q, R, &c. are functions of x, y, which gives place to a series
i integrations, introducing for each of them an arbitrary function.
One of the next easiest equations to integrate is this •
(af) = «•
which P and Q will always denote two functions of x and y.
Make
d
= U
and the proposed will transform to
To integrate this, we consider x constant, and then it contains only
two variables y and u, and it will be of the same form as the equation
dy + Pydx = Qdx
whose integral (see Vol. 1. p. 109) is
y = e-/*ax {/Qe/"«dx + CJ.
Hence our equation gives
u =e-
ANALYTICAL GEOMETRY.
But
U =
Hence by integration we get
z =f{ e-'pdy (/Qe'PJydy) + p x } dy + -^x.
By the same method we may integrate
-p. /d z\ ~ d2z -p, /d z\ ~
+ p (dx) = Q- a^u + p Civ) = Q>
K! y>
in which P, Q represent functions of x, and because of the divisor d x d y,
we perceive that the value of z will not contain arbitrary functions of the
same variable.
THE DETERMINATION OF THE ARBITRARY FUNCTIONS WHICH ENTEll
THE INTEGRALS OF EQUATIONS OF PARTIAL, DIFFERENCES O.Y
THE FIRST ORDEK.
The arbitrary functions which complete the integrals of equations of
partial differences, ought to be given by the conditions arising from the
nature of the problems from which originated these equations ; problems
generally belonging to the physical branches of the Mathematics.
But in order to keep in view the subject we are discussing, we shall
limit ourselves to considerations purely analytical, and we shall first seek
what are the conditions contained in the equation
/d z\ _
Vd x/
Since z is a function of x, y, this equation may be ;,msidered as that of
a surface. This surface, from the nature of its equation, has the followino-
property, that f-r— 1 must always be constant. Hence it follows that
every section of this surface made by a plane parallel to that of x, y is a
straight line. In fact, whatever may be the nature of this section, if we
divide it into an infinity of parts, these, to a small extent, may be con
sidered straight lines, and will represent the elements of the section, or.e
of these elements making with a parallel to the axis of abscissae, an angle
/dz\
whose tangent is (7-). Since this angle is constant, it follows that all
the angles formed in like manner by the elements of the curve, with par-
4 4
Ivi INTRODUCTION.
allels to the axis of abscissae will be equal. Which proves that the sec
tion in question is a straight line.
We might arrive at the same result by considering the integral of the
equation
= a
^u x/
which we know to be
z = a x + p y,
since for all the points .of the surface which in the cutting plane, the or-
dinate is equal to a constant c. Replacing therefore p y by p c, and
making p c = C, the above equation becomes
z = a x + C ;
this equation being that of a straight line, shows that the section is a
straight line.
The same holding good relatively to other cutting planes which may be
drawn parallel to that of x, z, we conclude that all these planes will cut the
surface in straight lines, which will be parallel, since they will each form
with a parallel to the axis of x, an angle whose tangent is a.
If, however, we make x = 0, the equation z = a x + p y reduces to
z =: Py» and will be that of a curve traced upon the plane of y, z; this
curve containing all the points of the surface whose coordinates are x = 0,
will meet the plane in a point whose coordinate is x =0; and since we
have also y = c, the third coordinate by means of the equation
z = ax + C
will be
z = C.
What has been said of this one plane, applies equally to all others
which are parallel to it, and it thence results that through all the points
of the curve whose equation is z = p y, and which is traced in the plane
of y, z, will pass straight lines parallel to the axis of x. This is ex
pressed by the equations
'd z>
/ z\
( -r- ) =
\d x/
and
Z = ax + py;
and since this condition is always fulfilled, whatever may be the figure of
the curve whose equation is z — <p y, we see that this curve .is arbi
trary.
From what precedes, it follows that the curve whose equation is z = py»
ANALYTICAL GEOMETRY.
Ivii
may be composed of arcs of different curves, which unite at their extre
mities, as in this diagram
or which have a break off in their course, as in this figure.
,N
In the first case the curve is discontinuous^ and in the second it is dis
contiguous. We may remark that in this last case, two different ordinates
P M, P N corresponding to the same abscissa A P; finally, it is possible,
that without being discontiguous, the curve may be composed of an in
finite series of arcs indefinitely small, which belong each of them to
different curves ; in this case, the curve is irregular, as will be, for
instance, the flourishes of the pen made at random ; «but in whatever way
it is formed, the curve whose equation is z = <p y, it will suffice, to con
struct the surface, to make a straight line move parallelly with this condi
tion, that its general point shall trace out the curve whose equation is
z =
and vhich is traced at random upon the plane of y, z.
If instead of the equation
/d z\
(di) = a'
we had
/d z\ Y
Id x) ~ X>
in which X was a function of x, then in drawing a plane parallel to the
plane (x, z), the surface will be cut by it no longer in a straight line, as
in the preceding case. In fact, for every point taken in this section, the
tangent of the angle formed by the element produced of the section, with
a parallel to the axis of x, will be equal to a function X of the abscissa x
of this point; and since the abscissa x is different for overy point :t foJ-
Iviii .. INTRODUCTION.
lows that this angle will be different at each point of the section, which
section, therefore, is no longer, as before, a straight line. The surface
will be constructed, as before, by moving the section parallelly, so that its
point may ride continually in the curve whose equation is z = <p y.
Suppose now that in the preceding equation, instead of X we have a
function, P of x, and of y. The equation
(T-Z) = p'
VI x/
containing three variables will belong still to a curve surface. If we cut
O .
this surface by a plane parallel to that of x, z, we shall have a section in
which y will be constant ; and since in all its points (j^) wi^ be eclual
to a function of the variable x, this section must be a curve, as in the pre
ceding case. The equation
(i-z) = P
VI x/
being integrated, we shall have for that of the surface
z =/Pdx + py;
if in this equation we give successively to y the increasing values y', y",
y'", &c. and make P, P', P"', &c. what the function P becomes in these
cases, we shall have the equations
z = /P'dx + ®y', z =/P"dx + py" 1
z = /P'"dx + py"', z = /P""dx + py"" &c. /
and we see that these equations will belong to curves of the same nature,
but different in form, since the values of the constant y will not be the
same. These curves are nothing else than the sections of the surface
made by planes parallel to the plane (x, z) ; and in meeting the plane
(y, z) they will form a curve whose equation will be obtained by equating
to zero, the value of x in that of the surface. Call the value of/Pdx,
in this case, Y, and we shall have
z = Y + py;
and we perceive that by reason of p y, the curve determined by this equa.-
tion must be arbitrary. Thus, having traced at pleasure a curve, Q R S,
upon the plane (y, z), if we represent by R L the section whose equation
Q
L
is z = f P'd x -f f> y', we shall move this section, always keeping the ex-
ANALYTICAL GEOMETRY. lix
tremity R applied to the curve Q R S ; but so that this section as it
moves, may assume the successive forms determined by the above group
of equations, and we shall thus construct the surface to which will belong
the equation
Era = R
Finally let us consider the general equation
whose integral is U = <p V. Since U = a, V = b, each of these equa«
tions subsisting between three coordinates, we may regard them as be
longing to two surfaces ; and since the coordinates are common, they
ought to belong to the curve of intersection of the two surfaces. This
being shown, a and b being arbitrary constants, if in U = a, we give to
X and y the values x', y' we shall obtain for z, a function of x', of y' and
of a, which will determine a point of the surface whose equation is U = a.
This point, which is any whatever, will vary in position if we give succes
sively different values to the arbitrary constant a, which amounts to say
ing that by making a vary, we shall pass the surface whose equation is
U = a, through a new system of points. This applies equally to V = b,
and we conclude that the curve of intersection of the two surfaces will
change continually in position, and consequently will describe a curved
surface in which a, b may be considered as two coordinates ; and since
the relation a = <p b which connects these two coordinates, is arbitrary
we perceive that the determination of the function <p amounts to making
a surface pass through a curve traced arbitrarily.
To show how this sort of problems may conduct to analytical condi
tions, let us examine what is the surface whose equation is
d z\ /d
= x
We have seen that this equation being integrated gives
z = p(x2 + y1).
Reciprocally we hence derive
x* + yj = 0>z.
If we cut the surface by a plane parallel to the plane (x, y) the equation
of the section will be
x2 + y* = <& c;
and representing by a * the constant 4> c, we shall have
x1 + y2 = a2.
This equation belongs to the circle. Consequently the surface will
U
INTRODUCTION.
have this property, viz. that every section made by a plane parallel to the
plane (x, y) will be a circle.
This property is also indicated by the equation
d
for this equation gives
dy
x = y -J-&.
J d x
This equation shows us that the subnormal ought to be always equal to
the abscissa which is the property of the circle.
The equation z = <p (x2 + y*) showing merely that all the sections
parallel to the plane (x, y) are circles, it follows thence that the law ac
cording to which the radii of these sections ought to increase, is not
comprised in this equation, and that consequently, every surface of revo
lution will satisfy the problem ; for we know that in this sort of surfaces,
the sections parallel to the plane (x, y) are always circles, and it is need
less to say that the generatrix which, during a revolution, describes the
surface, may be a curve discontinued, discontiguous, regular or irregular.
Let us therefore investigate the surface for which this generatrix will
be a parabola A N, and suppose that, in this hypothesis, the surface is
cut by a plane A B, which shall pass through the axis of z ', the trace of
B
Q
this plane upon the plane (x, y) will be a straight line A L, which, being
drawn through the origin, will have the equation y = a x ; if we repre
sent by t the hypothenuse of the right angled triangle A P Q, constructed
upon the plane (x, y) we shall have
tj = x2 + yai
but t being the abscissa of the parabola A M, of which Q M = 2 i» the
ordinate, we have, by the nature of the curve,
t* = bz.
Putting for t 2 its value x * + y !, we get
Z= (y •
orz
= ^x«(» + n');
ANALYTICAL GEOMETRY. ixi
- and making
i (a» + 1) = m,
we shall obtain
z = mx2;
so that the condition prescribed in the hypothesis, where the generatrix
is a parabola, is that we ought to have
z = m x *, when y = a x.
Let us now investigate, by means of these conditions, the arbitrary
function which enters the equation z = <f> (x! + y *). For that pur
pose, we shall represent by U the quantity x * + y 3» which is effected by
the symbol <p, and the equation then becomes
z = f> U;
and we shall have the three equations
x * + y * = U, yrrax, z = m x *.
By means of the two first we eliminate y and obtain the value of x *
which being put into the third, will give
Z = m . ^ — - - r
1 + a* >
an equation which reduces to
7, — TT •
• b" U>
the value of z being substituted in the equation z = <f> U, will change
it to
and putting the value of U in this equation, we shall find that
and we see that the function is determined. Substituting this value of
<p (x z + y 9) in the equation z — <f> (x2 + y2), we get
Z= b(x* + y2)'
for the integral sought, an equation which has the property required,
since the hypothesis of y = ax gives
z = m x *.
This process is general ; for, supposing the conditions which determine
the arbitrary constant to be that the integral gives F (x, y, z} = 0, when
we have f (x, y, z) = 0, we shall obtain a third equation by equating to
ixii
INTRODUCTION.
CJ the quantity which follows p, and then by eliminating, successively,
two of the variables x, y, z, we shall obtain each of these variables in a
function of U ; putting these values in the integral, we shall get an equa
tion whose first 'member is <p U, and whose second member is a compound
expression in terms of U ; restoring the value of U in terms of the vari-
bles, the arbitrary function will be determined.
THE ARBITRARY FUNCTIONS WHICH ENTER THE INTEGRALS OF THE
EQUATIONS OF PARTIAL DIFFERENCES OF THE SECOND ORDER.
Equations of partial differences of the second order conduct to integrals
which contain two arbitrary functions ; the determination of these func
tions amounts to making the surface pass through two curves which may
be discontinuous or discontiguous. For example, take the equation
•whose integral has been found to be
Let A x, A y, A z, be the axis of coordinates; if we draw a plane
K L parallel to the plane (x, z), the section of the surface by this plane
will be a straight line ; since, for all the points of this section, y being
equal to A p, if we represent A p by a constant c, the quantities <p y, ^ y
will become <p c, -^ c, and, consequently, may be replaced by two con
stants, a, b, so that the equation
z = x py -f 4y
ANALYTICAL GEOMETRY. Ixiii
will become
z = a x -f- b,
and this is the equation to the section made by the plane K L.
To find the point where this section meets the plane (y, z) make
x = 0, and the equation above gives z = -4/ y, which indicates a curve
a m b, traced upon the plane (y, z). It will be easy to show that the
section meets the curve a m b in a point m ; and since this section is a
straight line, it is only requisite, to find the position of it, to find a second
point. For that purpose, observe that when x = 0, the first equation
reduces to
z = ^y,
whilst, when x = 1, the same equation reduces to
z = 9 y + 4- y-
Making, as above, y = Ap = c, these two values of z will become
z = b, z = a -f b,
and determining two points m and r, taken upon the same section, in r
we know to be in a straight line. To construct these points we thus pro
ceed : we shall arbitrarily trace upon the plane (y, z) the curve a m b,
and through the point p, where the cutting plane K L meets the axis of
y, raise the perpendicular pm = b, which will be an ordinate to the
curve ; we shall then take at the intersection H L of the cutting plane,
and the plane (x, y), the part p p' equal to unity, and through the point
p', we shall draw a plane parallel to the plane (y, z), and in this plane
construct the curve a' m' b', after the modulus of the curve a m b, and so
as to be similarly disposed ; then the ordinate m' p' will be equal to m p ;
and if we produce m' p' by m' r, which will represent a, we shall deter
mine the point r of the section.
If, by a second process, we then produce all the ordinates of the curve
a' m' b', we shall construct a new curve a' r' b', which will be such, that
drawing through this curve and through a m b, a plane parallel to the
plane (x, z), the two points where the curves meet, will belong to the
same section of the surface.
From what precedes, it follows that the surface may be constructed, by
moving the straight line m r so as continually to touch the two curves,
a m b, a' m' b'.
This example suffices to show that the determination of the arbitrary
functions which complete the integrals of equations of partial differences
of the second order, is the same as making the surface pass through two
curves, which, as well as the functions themselves, may be discontinuous,
discontiguous, regular or irreguiar.
Ixir INTRODUCTION
CALCULUS OF VARIATIONS.
If we have given a function Z — F, (x, y, y', y"), wherein y', y" mean
y itself being a function of x, it may be required to make L have certain
properties, (such as that of being a maximum, for instance) whether by
assigning to x, y numerical values, or by establishing relations between
these variables, and connecting them by equations. When the equation
y = p x is given, we may then deduce y, y7, y" . . . in terms of x and sub
stituting, -we have the form
Z = f x.
By the known rules of the differential calculus, we may assign the values
ofx, when we make of x a maximum or minimum. Thus we determine what
are the points of a given curve, for which the proposed function Z, is
greater or less than for every other point of the same curve.
But if the equation y = <p x is not given, then taking successively for
<f> x different forms, the function Z = f x will, at the same time, assume
different functions of x. It may be proposed to assign to f x such a
form as shall make Z greater or less than every other form of p ^for the
same numerical value ofx 'whatever it may be in other respects. This latter
species of problem belongs to the calculus of variations. This theory
relates not to maxima and minima only; but we shall confine our
selves to these considerations, because it will suffice to make known all
the rules of the calculus. We must always bear in mind, that the varia
bles x, y are not independent, but that the equation y = px is unknown,
and that we only suppose it given to facilitate the resolution of the prob
lem. We must consider x as any quantity whatever which remains the same
for all the differential forms of <p x ; the forms of <p, p', <p" . . . . are therefore
variable, whilst x is constant.
In Z = F (x, y, y', y". . .) put y + k for y, y' + k', for y'. . . , k being
an arbitrary function of x, and k', k./' . . . the quantities
dl^ dMc
dV dx*"
But, Z will become
Z, = F (X, y -f k, y' + k', y" + k," . . .)
ANALYTICAL GEOMETRY. Ixv
Taylor's theorem holds good whether the quantities x, y, k be depen
dent or independent. Hence we have
so that we may consider x, y, y', y" . . . as so many independent variables.
The nature of the question requires that the equation y = <p x should
he determined, so that for the same value of x, we may have always
Z7 > Z, or Z/ < Z : reasoning as in the ordinary maxima and minima,
we perceive that the terms of the first order must equal zero, or that we
have
Since k is arbitrary for every value of x, and it is not necessary that its
value or its form should remain the same, when x varies or is constant,
k', k" . . . is as well arbitrary as k. For we may suppose for any value
x = X that k = a + b (x — X) + | c (x — X) * + &c., X, a, b, c . . .
being taken at pleasure ; and since this equation, and its differentials,
ought to hold good, whatever is x, they ought also to subsist when
x = X, which gives k = a, k' = b, k" = c, &c. Hence the equation
Z, = Z + . . . cannot be satisfied when a, b, c . . . are considered inde
pendent, unless (see 6, vol. I.)
/d Zx /d Zx /d ZN / d Z
3p = °' (ay) == °' (37') = °v- (dyrn
n being the highest order of y in Z. These different equations subsist
simultaneously, whatever may be the value of x ; and if so, there ought
to be a maximum or minimum ; and the relation which then subsists be
tween x, y will be the equation sought, viz. y = <p x, which will have the
property of making Z greater or less than every other relation between
x and y can make it. We can distinguish the maximum from the mini
mum from the signs of the terms of the second order, as in vol. I.
p. (31.)
But if all these equations give different relations between x, y, the
problem will be impossible in the state of generality which we have
ascribed to it ; and if it happen that some only of these equations subsist
mutually, then the function Z will have maxima and minima, relative to
some of the quantities y, y', y" . . . without their being common to them
all. The equations which thus subsist, will give the relative maxima and
minima. And if we wish to make X a maximum or minimum only relatively
ixvi INTRODUCTION.
to one of the quantities y, y', y" . . . , since then we have only one equa
tion to satisfy, the problem will be always possible.
From the preceding considerations it follows, that first, the quantities
X, y depend upon one another, and that, nevertheless, we ought to make
them vary, as if they were independent, for this is but an artifice to get
the more readily at the result.
Secondly, that these variations are not indefinitely small ; and if we em
ploy the differential calculus to obtain them, it is only an expeditious
means of getting the second term o£ the developement, the only one
which is here necessary.
Let us apply these general notions to some examples.
Ex. 1. Take, upon the axis of x of a curve, two abscissas m, n; and
draw indefinite parallels to the axis of y. Let y = <p x be the equation
of this curve: if through any point whatever, we draw a tangent, it will
cut the parallels* in points whose ordinates are
1 = y + y' (m — x), h = y + y' (n — x) .
If the form of 9 is given, every thing else is known; but if it is not
given, it may be asked, what is the curve which has the property of
having for each point of tangency, the product of these two ordinates less
than for every other curve.
Here we have 1 X h ; or
Z = { y X (m — x) y' } + { y + (n - x) y' J .
From the enunciation of the problem, the curves which pass through the
same point (x, y) have tangents taking different directions, and that which
is required, ought to have a tangent, such that the condition Z = maximum
is fulfilled. We may consider x and y constant ; whence
/d_Z\ • . 2y' _ 2 x — in — n 1 1
V d yv " ' y ~ (x — m) (x — n) ~ x — mx — n*
Then integrating we get
y2 = C(x — m) (x — n).
The curve is an ellipse or a hyperbola, according as C is positive or
negative ; the vertexes are given by x = m, x =s n ; in the first case, the
product h X 1 or Z is a maximum, because y" is negative; in the second,
Z is a minimum or rather a negative maximum ; this product is moreover
constant, and 1 h = — 1 C (m — n) 2, the square of the semi-axis.
Ex. 2. What is the curve for which, in each of its points, the square of
the subnormal added to the abscissa is a minimum ?
We have in this case
Z = (y y + x) 2
ANALYTICAL GEOMETRY. Ixvii
whence \ve get two equations subsisting mutually by making
y y' + x = 0
and thence
x 2 + y 2 = r \
Therefore all the circles described from the origin as a center wi" 1 alone
satisfy the equation.
The theory just expounded has not been greatly extended ; but it serves
as a preliminary developement of great use for the comprehension of a
far more interesting problem which remains to be considered. This re
quires all the preceding reasonings to be applied to a function of the form
/* Z: the sign y indicates the function Z to be a differential and that after
having integrated it between prescribed limits" it is required i,o endow it
with the preceding properties. The difficulty here to be overcome is that
of resolving the problem without integrating.
When a body is in motion, we may compare together either the differ
ent points of the body in one of its positions or the plane occupied suc
cessively by a given point. In the first case, the body is considered fixed,
and the symbol d will relate to the change of the coordinates of its surface;
in the second, we must express by a convenient symbol, variations alto
gether independent of the first, which shall be denoted by 8. When we
consider a curve immoveabie, or even variable, but taken in one of its po
sitions, d x, d y . . . announce a comparison between its coordinates ; but
to consider the different planes which the same point of a curve occupies,
the curve varying in form according to any law whatever, we shall write d
x, 5 y ... which denote the increments considered under this point of view,
and are functions of x, y . . . In like manner, d x becoming d (x + <3 x)
will increase by d 5 x ; d 2 x will increase by d 2 «3 x, &c.
Observe that the variations indicated by the symbol <3 are finite, and
wholly independent of those which d represents ; the operations to which
these symbols relate being equally independent, the order in which they
are used must be equally a matter of indifference as to the result. So
that we have
<5.d x = d. 5 x
d 2. 8 x = 3 . d 2 x
&c.
/a U = * -• U.
and so on.
It remains to establish relations between x, y, 7. . .such that/Z may
be a maximum or a minimum letween given limits. That the calculus may
he rendered the more symmetrical, we shall not suppose any differential
Ixviu INTRODUCTION
constant ; moreover we shall only introduce three variables because it will
be easy to generalise the result. To abridge the labour of the process,
make
d x = x/5 d 2 x — x//5 &c.
so that
z = F (x, x,, x/y, . . . y, yy, ylfl . . . z, z,, z,, . . .).
Now x, y and z receiving the arbitrary and finite increments 3 x, d y,
8 z, d x or x, becomes
d (x + a x) = d x + a d x or x, + 6 x,.
In the same manner, x,, increases by a x,, and so on ; so that develop
ing Z, by Taylor's theorem, and integrating / Z becomes
The condition of a maximum or minimum requires the integral of the
terms of the first order to be zero between given limits whatever may be
ii x, B y, d z as we have already seen. Take the differential of the known
function Z considering x, x/} X// . . . y, y,, y,, ... as so many independent
variables ; we shall have
dZ = mdx4-ndx + p d x + . . . M d y + N d yx. . . + /A d z + v d z/ . . .
m n ... M, N .../*, v ... being the coefficients of the partial differences
of Z relatively to x, x7 . . . y, y, . . . z, z/} . . . considered as so many varia
bles ; these are therefore known functions for each proposed value of Z.
' Performing this differentiation exactly in the same manner by the symbol
3, we have
But this known quantity, whose number of terms is limited, is precisely
that which is under the sign /, in the terms of the first order of the de-
velopement : so that the required condition of max. or min. is that
/3Z = 0,
between given limits, whatever may be the variations 5 x, 8 y, d z. Ob
serve, that here, as before, the differential calculus is only employed as a
means of obtaining easily the assemblage of terms to be equated to zero ;
so that the variations are still any whatever and finite.
ANALYTICAL GEOMETRY.
We have said that d . 8 x may be put for d . 3 x ; thus the first line is
equivalent to
m, n . . . contains differentials, so that the defect of homogeneity is here
only apparent. To integrate this, we shall see that it is necessary to
disengage from the symbol f as often as possible, the terms which con
tain d 3. To effect this, we integrate by parts which gives
y n d 3 x = n . 3x — yd n . 3 x
/p.d23x = p d 3x — d p 3x+/d'p3x
yqd35x = qd23x — dq.d3x-f- d2q.dx — f d 3 q . 3 x
&c.
Collecting these results, we have this series, the law of which is easily
recognised ; viz.
/ (m — d n + d * p — d 3 q + d 4 r — . . .)3x
-f- (p — d q + d 2 r — d 3 s -f d 4 t — . . .) d 3 x
+ (q — d r + . . .) d 2 3 x
+ &c.
The integral of (A) ory. 3 z = 0 , becomes therefore
i- d n +d 8 p-...)3 x + (M-d N+d 2 P-...)3 y+ (//-d v-...)8 z] =0
C
J
(.
+ (q-dr...) d23x ...+ K = 0
K being the arbitrary constant. The equation has been split into two,
because the terms which remain under the sign y cannot be integrated, at
least whilst 3 x, 3 y, 8 z are arbitrary. In the same manner, if the nature
of the question does not establish some relation between 3 x, 3 y, 3 z, the
independence of these variations requires also that equation (B) shall again
make three others ; viz.
0=m — dn + d 2 p — djq -f- d 4 r — • • • 1
0 = M — dN+d2P — d3Q+d4R— .... S- . . (D)
Consequently, to find the relations between x, y, z, which make y Z a
maximum, we must take the differential of the given function Z by con
sidering x, y, z, d x, d y, d z, d z x, . . . as so many independent vari
ables, and use the letter 3 to signify their increase; this is what is termed
taking the variation of Z. Comparing the result with the equation (A),
we shall observe the values of m, M, /i, n, N ... in terms of x, y, z, and
e3
LXK INTRODUCTION.
their differences expressed by d. We must then substitute these in the
equations (C), (D) ; the first refers to the limits between which the
maximum should subsist ; the equations (D) constitute the relations re
quired; they are the differentials of x, y, z, and, excepting a case of
absurdity, may form distinct conditions, since they will determine nume
rical values for the variables. If the question proposed relate to Geo
metry, these equations are those of a curve or of a surface, to which
belongs the required property.
As the integration is effected and should be taken between given limits,
the terms which remain and compose the equation (C) belong to these
limits : it is become of the form K + L = 0, L being a function of
x, y, z, 8 x, 8 y, d z . . . Mark with one and two accents the numerical
values of these variables at the first and second limit. Then, since the
integral is to be taken between these limits, we must mark the different
terms of L which compose the equation C, first with one, and then with
two accents ; take the first result from the second and equate the differ
ence to zero ; so that the equation
L/x - L, = 0
contains no variables, because x, d x . . . will have taken the values
x/} 3 x/ . . . x//5 o x7/ . . . assigned by the limits of the integration. We
must remember that these accents merely belong to the limits of the
integral.
There are to be considered four separate cases.
1. If the limits are given andjixed, that is to say, if the extreme values
of x, y, z are constant, since a x,, d 8 x, . . . d x,,, d 8 x/x, &c. are zero, all
the terms of L, and L,, are zero, and the equation (C) is satisfied. Thus
we determine the constants which integration introduces into the equations
(D), by the conditions conferred by the limits.
2. If the limits are arbitrary and independent, then each of the coeffi
cients a x, , 3 x/y . . . in the equation (C) is zero in particular.
3. If there exist equations of condition, (which signifies geometrically
that the curve required is terminated at points which are not fixed, but
which are situated upon two given curves or surfaces,) for the limits, that
is to say, if the nature of the question connects together by equations,
some of the quantities x,, y/5 z/} x//} y,,, z/7 we use the differentials of these
equations to obtain more variations d x,, 3 y,, 3 z,, d x;/, &c. in functions
of the others; substituting in L/7 — L, = 0, these variations will be re-
duced to the least number possible : the last being absolutely independent,
the equation will split again into many others by equating separately their
coefficients to zero.
ANALYTICAL GEOMETRY. Ixxi
Instead of this process, we may adopt the following one, which is more
elegant. Let
u = 0, v = 0, &c.
be the given equations of condition ; we shall multiply their variations
t u, 3 v ... by the indeterminates X, X'. . . This will give Xdu + X'Sv + ...
a known function of d x/} 6 x//5 d y, . . . Adding this sum to Lx/ — L,, we
shall get
L,, — L, + X d u + X' d v + . . . = 0 . . . . (E).
Consider all the variations 8 x/} d x//} ... as independent, and equate
their coefficients separately to zero. Then we shall eliminate the inde
terminates X, X'. . . from these equations. By this process, we shall arrive
at the same result as by the former one ; for we have only made legiti
mate operations, and we shall obtain the same number of final equations.
It must be observed, that we are not to conclude from u = 0, v = 0,
that at the limits we have d u = 0, d v = 0 ; these conditions are inde
pendent, and may easily not coexist. In the contrary case, we must
consider d u = 0, d v = 0, as new conditions, and besides X d u, we
must also take X' d d u . . .
4. Nothino- need be said as to the case where one of the limits is fixed
O
and the other subject to certain conditions, or even altogether arbitrary,
because it is included in the three preceding ones.
It may happen also that the nature of the question subjects the varia
tions o X, d y, d z, to certain conditions, given by the equations
i - 0, 6 = 0,
and independently of limits; thus, for example, when the required curve
is to be traced upon a given curve surface. Then the equation (B) will
not split into three equations, and the equations (D) will not subsist. We.
must first reduce, as follows, the variations to the smallest number possi
ble in the formula (B), by means of the equations of condition, and equate
to zero the coefficients of the variations that remain ; or, which is tanta
mount, add to (B) the terms X5? + X'60 + ...; then split this equation
into others by considering d x, 6 y, 3 z as independent ; and finally elimi
nate X, X' ...
It must be observed, that, in particular cases, it is often preferable to
make, upon the given function Z, all the operations which have produced
the equations (B), (C) instead of comparing each particular case with the
general formulae above given.
Such are the general principles of the calculus of variations: let us
illustrate it with examples.
Ixxii
INTRODUCTION.
Ex. 1. What is the curve C M K of which the length M K, comprised
between the given radii-vectors A M, A K is the least possible.
We have, (vol. I, p. Q00)> if r be the radius-vector,
s = /(r!d<?* + d 2) = Z
it is required to find the relation r = <p 6, which j-end&rs Z a minimum
the variation is
7 _ r d <?2. a .f r2d 4. ad d + dr . od r
V (r * d 6 • + d r «)
Comparing with equation (A); where we suppose x = r, y = 6t we
have
r d 6 z d r r * d d
m = — j — , n = —. — , M = 0 , N = — ,
as d s d s
the equations (D) are
r d 6*
ds
T
d s
_
— C»
Eliminating d 0, and then d s, from these equations, and d s 2 = r * d P;
4- d r 2, we perceive that they subsist mutually or agree; so that it is
sufficient to integrate one of them. But the perpendicular A I let fall
from the origin A upon any tangent whatever. T M is
A J = A M + sin. A M T = r sin. /?,
which is equivalent, as we easily find, to
r tan. 0
which gives
V (1 + tan. 2 /3)
d 6
V (r * d 6* + d r ') ' ~dl~
and since this perpendicular is here constant, the required line is a
straight line. The limits M and K being indeterminate, the equations
(C) are unnecessary.
Ex. 2. To Jind the shortest line between two given points, or two given
curves.
ANALYTICAL GEOMETRY. Ixxiii
The length" s of the line is
/Z =fV (dx2 + dy* + dz2).
It is required to make this quantity a minimum ; we have
, ,7 dx, , d y . , dz..
o L •=. -, — a d x + -r-^ d y + -5 — <5 d z,
d s d s d s
ind comparing with the formula (A), we find
rti\T d x XT d y dz
m = 0,M = 0,/A = 0,n= — ,— , N = ~^-- , v sr -^— :
as as d s
the other coefficients P, p, * . . . are zero. The equations (D) become*
therefore, in this case,
whence, by integrating
Squaring and adding, we get
a«+ b2 + c2 = 1,
a condition that the constants a, b, c must fulfil in order that these equa
tions may simultaneously subsist. By division, we find
d y _ b d z _ c_
d x ~~ a' oTx ~ a*
whence
b x = a y + a', c x = a z + b' ;
the projections of the line required are therefore straight lines — the line is
therefore itself a straight line.
To find the position of it, we must know the five constants a, b, c,
a', b'. If it be required to find the shortest distance between two given
fixed points (x , y,, z,), (XA, y//} zj, it is evident that a, x, a x/7, ay,... are
zero, and that the equation (C) then holds good. Subjecting our two
equations to the condition of being satisfied when we substitute therein
x/» x/,» y/ • • • f°r x/ y/ z, we shall obtain four equations, which, with
a 2 + b a + c 2 = 1, determine the five necessary constants.
Suppose that the second limit is a fixed point (x//? y//} z/7), in the plane
(x, y), and the first a curve passing through the point (x/5 yy z;), and also
situated in this plane ; the equation
b x =r a y + a'
then suffices. Let y, = f x, be the equation of the curve ; hence
ay/ = A3X/;
the equation (C) becomes
Ixxiv INTRODUCTION.
and since the second limit is fixed it is sufficient to combine together the
equations
dy, = ASx,
dx,3x, + dy/5y/ =r 0.
Eliminating d y, we get
dx, + Ady, = 0.
We might also have multiplied the equation of condition
S y, — A S x, = 0
by the indeterminate X, and have added the result to L,, which would
have given
(a-f) 8x' + On) Sy' + ^y<-xASx< = 0>
whence
_ x A = 0, (^ ) + x = 0.
d s
Eliminating X we get
dx, + Ady/ = 0.
But then the point (x/} y,) is upon the straight line passing through the
points (X/, y/, Z/), {x//$ y//, Z//), and we have also
b d x, = ad y/}
whence
a = — b A
and
ly = - -1 = -;
dx A a
which shows the straight line is a normal to the curve of condition. The
constant a' is determined by the consideration of the second limit which is
given and fixed.
It would be easy to apply the preceding reasoning to three dimensions,
and we should arrive at similar conclusions; we may, therefore, infer
generally that the shortest distance between two curves is the straight
line which is a normal to them.
If the shortest line required were to be traced upon a curve surface
whose equation is u - 0, then the equation (B) would not decompose into
three others. We must add to it the term X d u ; then regarding 6 x, 5 y,
fi z as independent, we shall find the relations
ANALYTICAL GEOMETRY.
Ixxv
*
From these eliminating A, we have the two equations
d u\ dxv du\ , /d
/d u\ , /xv /u
(dz)d.(dl)= (<Tx
d z\ /du\t
\vhich are those of the curve required.
Take for example, the least distance measured upon the surface of a
1C'
sphere, whose center is at the origin of coordinates : hence
u = x,2 + y« + z* — r2 = 0»
=2x, =
^d y/
Our equations give, making d s constant,
whence
y d z x = x d * y.
Integrating we have
zdx — xdz = ads, zdy — ydz = bds, ydx — xdy = cds.
Multiplying the first of these equations by — y, the second by x, the
third by z, and adding them, we get
ay = bx + cz
the equation of a plane passing through the origin of coordinates. Hencf
the curve required is a great circle which passes through the points A'
C', or which is normal to the two curves A' B and C' D which are limits
and are given upon the spherical surface.
When a body moves in a fluid it encounters a resistance which ceteris
Ixxvi INTRODUCTION.
paribus depends on its form (see vol. I.) : if the body be one of revolu
tion and moves in the direction of its axis, we can show by mechanics
that the resistance is the least possible when the equation of the gener
ating curve fulfils the condition
/y d d y3
* , . J ' — , = minimum,
d x* + d y2
or
1+ y'2 '
Let us determine the generating curve of the solid of least resistance
(see Principia, vol. II.).
Taking the variation of the above expression, we get
. y/3dx
(i+y'2)s
the second equation (D) is
M — dN = 0;
and it follows from what we have done relatively to Z, that
= y'dN+ Nd/,
&c
'
because
M = d N.
Thus integrating, we have
'3
.4.XXLZ- - N v ~ y_y_^_±_o .
'14. y~'« ~ L y (1 + y'2)*
Therefore
a (I + y/2)2 = 2yy'3.
Observe that the first of the equations (D) or m — d n = 0, would
have given the same result — n = a ; so that these two equations conduct
to the same result. We have
a (1 +_/^)J
y- --^y> '
substituting for y its value, this integral may easily be obtained ; it remains
to eliminate f from these values of x and y, and we shall obtain the
equation of the required curve, containing two constants which we shall
determine from the given conditions.
ANALYTICAL GEOMETRY. Ixxvii
Ex. 3. WJiat is the curve ABM in which the area B O D M comprised
between the arc B M the radii of curvature B O, D M and the arc O D
of the evolute, is a minimum ?
The element of the arc A M is
dsrrdxvM-fy ;
the radius of curvature M D is
and their product is the element of the proposed area, or
^
y" d x d y
It is required to find the equation y = f x, which makes f Z, a mini
mum.
Take the variation d N, and consider only the second of the equations
(D), which is sufficient for our object, and we get
M = 0, N — d P = 4 a,
XT dx* + dy2 1 + y'*
N = — -, — , , J . 4 d y = — ,/ 4 y',
d x d * v J v" J '
P_
y'/2dx
But
V'
y"d x
= 4 a d y + d P d y' -f. P d y" d x,
putting 4 a + P for N. Moreover y" d x = d y', changes the last
terms into
(y" d P -f P d y") d x = d (P y").. d x = — d
Ixxviii INTRODUCTION.
Integrating, therefore,
/I _I_ i.' 2\ 2
= a yf + b,
2y
_dy' d 2(ay' + b)dy'
— A „ » U A — /I I „/ !!\ 2 '
~2(ay' + b) ~dx' (1 + y")
finally,
On the other side we have
y =yy d x = y' x — /x d y'
or
y = y' x — c y' — f^^* d y' — /b d y tan.-1/;
this last term integrates by parts, and we have
y i= y7 x — c y' — (by — a) tan.-'y + f.
Eliminating the tangent from these values of x and y, we get
by = a(x-~c) + (bf ~ya)/ + bf,
(by — a) d x b d y — a d x
V(by-ax+g)=i -gj- -,d.= V(bylax + g);
finally,
s = 2 V (b y — a x -f g) + h.
This equation shows that the curve required is a cycloid, whose four
constants will be determined from the same number of conditions.
Ex. 4. What is the curve of a given length s, between two fixed points,
for which f y d s is a maximum ?
We easily find
. /d x\ , c d y
(V + ^) ( -i— ) = c , whence d x = • , c , - ; — Hf-t - ^ ',
u ' \d s/ V £ (y + X) * — c2]
and it will be found that the curve required is a catenary.
* is the vertical ordinate of the center of gravity of an arc
whose length is s, we see that the center of gravity of any arc whatever of
the catenary is lower than that of any other curve terminated by the
same points.
Ex. 5. Reasoning in the same way for f y * d x = minimum, and
J y d x = const, we find y * + X y = c, or rather y = c. We have
here a straight line parallel to x. Since '—^ — , — is the vertical ordinate
2/y dx
of the center of gravity of every plane area, that of a rectangle, whose
side is horizontal, is the lowest possible ; so that every mass of water
ANALYTICAL GEOMETRY. Ixxix
whose upper surface is horizontal, has its center of gravity the lowest
possible.
FINITE DIFFERENCES.
If we have given a series a, b, c, d, . . . take each term of it from that
which immediately follows it, and we shall form ihejirst differences, viz.
a' = b — a, b' = c — b, c' = d — c, &c.
In the same manner we find that this series a', b', c', d' . . . gives the
second differences
a" = b' — a', b" = c' — b', c" = d' — . c', &c.
which again give the third differences
a'" = b" — a", b'" = c" — b", c'" = d" — c", &c.
These differences are indicated by A, and an exponent being given to
it will denote the order of differences. Thus An is a term of the series
of nth differences. Moreover we give to each difference, the si<m which
belongs to it ; this is — , when we take it from a decreasing series.
For example, the function
y = x ' — 9x + 6
in making x successively equal to 0, 1, 2, 3, 4 ... gives a series of
numbers of which y is the general term, and from which we get the
following differences,
for x = 0, 1, 2, 3, 4, 5, 6, 7 ...
series y = 6, — 2, — 4, 6, 34, 86, 168, 286 . . .
first diff. A y = — 8, —2, 10, 28, 52, 82, 118 ...
second diff. A * y = 6, 12, 18, 24, 30, 36 ...
third diff. A 3 y = 6, 6, 6, 6, 6, ...
We perceive that the third differences are here constant, and that the
second difference is an arithmetic progression : we shall always arrive at
constant differences, whenever y is a rational and integer function of x ;
which we now demonstrate.
In the monomial k x m make x = a, j8, y, . . . 6, *, x (these numbers
having h for a constant difference), and we get the series
k « m, k /3 m, . . . k 6 m, k K ra, k X m.
Since K = X — h, by developing k x m — k (X — h) m, and designating
DV m, A', A" ... the coefficients of the binomial, we find, that
k (\ « _ x ") = k m h x » -1 — k A' h * X m~2 + k A" 3 h. . .
Ixxx INTRODUCTION.
Such is the first difference of any two terms whatever of the series
k « m, k /3 m . . . k x m, &c,
The difference which precedes it, or k (%, m — 6 m) is deduced by
changing X into x and x into 6' and since x = X — h, we must put
X — h for X in the second member :
k m h(X-h) "*-i-k A' h 2 (X-h m~2) ... = k m h X **-*-{ A/ + m(m-l)Jkh« X m~2 ...
Subtracting these differences, the two first terms will disappear, and
we get for the second difference of an arbitrary rank
k m (m — l)h2xm-2 + kB'h3xm-3 + ...
In like manner, changing X into X — h, in this last developement, and
subtracting, the two first terms disappear, and we have for the third
difference
km (m — 1) (m — 2) h'X"-3 -f k B" h 4Xm-*. . . ,
and so on continually.
Each of these differences has one term at least, in its developement,
like the one above ; the first has m terms ; the second has m — 1 terms;
third, m — 2 terms ; and so on. From the form of the first term, which
ends by remaining alone in the mth difference, we see this is reduced to
the constant
1 . 2 . 3 . . . in k h m.
If in the functions M and N we take for x two numbers which give the
results m, n ; then M + N becomes m + n. In the same manner, let
m', n' be the results given by two other values of x ; the first difference,
arising from M -f N, is evidently
(m — m') -f (n — n').
that is, the difference of the sum is the sum of the differences. The same
may be shown of the 3d and 4th . . . differences.
Therefore, if we make
x = a> & 7 ...
in
k x m + p x m~1 + . . .
the mth difference will be the same as if these were only the first term
k x m, for that of p x m— *, q x m~ 2 ... is nothing. Therefore the mth
difference is constant, lichen for x ive substitute numbers in arithmetic pro
gression, in a rational and integer function o/*x.
We perceive, therefore, that if it be required to substitute numbers in
arithmetic progression, as is the case in the resolution of numerical equa
tions, according to Newton's Method of Divisors, it will suffice to find
the (m + 1) first results, to form the first, second, &c. differences. The
ANALYTICAL GEOMETRY. Ixxxi
mlh difference will have but one term ; as we know it is constant and
= 1 . 2 . 3 . . . m k h Ir, we can extend the series at pleasure. That of
the (m — l)th differences will then be extended to that of two known
terms, since it is an arithmetic procession ; that of the (m — 2)th differ
ences will, in its turn, be extended ; and so on of the rest.
This is perceptible in the preceding example, and also in this; viz.
x =
0
. 1
.2.
3
3d Diff. 6 .
6 .
6
. 6
. 6 .
6
Series
1
.-1
.1.
13
2nd . . 4 .
10.
16
. 22
. 28 .
34
1st.
...
-2
.2.
12
1st . —2 .
2 .
12
. 28
. 50.
78
2nd
. .
3.
10
Results 1 .
j
1
. 13
. 41 .
91
3d
...
6
For x 0 .
1 .
2
. 3
. 4 .
5
These series are deduced from that which is constant
6.6.6.6...
and from the initial term already found for each of them : any term is
derived by adding the two terms on the left which immediately precede it.
They may also be continued in the contrary direction, in order to obtain
the results of x r= — 1, — 2, — 3, &c.
In resolving an equation it is not necessary to make the series of results
extend farther than the term where we ought only to meet with numbers
of the same sign, which is the case when all the terms of any column are
positive on the right, and alternate in the opposite direction ; for the
additions and subtractions by which the series are extended as required,
preserve constantly the same signs in the results. We learn, therefore,
by this method, the limits of the roots of an equation, whether they be
positive or negative.
Let yx denote the function of x which is the general term, viz. the
x + 1th, of a proposed series
y0 y2 + yi + . . . yx +
which is formed by making
x — 0, 1, 2, 3 ...
For example, y5 will designate that x has been made — 5, or, with re
gard to the place of the terms, that there are 5 before it (in the last ex
ample this is 91). Then
yi — yo = A yo» y* — yi = A yi » ya — ys = ys • • •
Ay' — Ay0 = A2y0, Ay*— A*yi = A^y, , A y3 _ A y2 = A*y8 . . .
A2y, — A*y0 = A3y0 , A*y2 — A*y, = A'y, , A*y3 — A*y2 = A3y2 . . .
&C.
Ixxxii . INTRODUCTION.
nix] generally we have
yx— yx_i = Ay x_!
Ay x — Ayx _ j = A-y x _ i
A2yx — A*yx_, = A3yx_!
&c.
Now let us form the differences of any series a, b, r, d . . . in this
manner. Make
. b = c + a'
c = b + b'
d = c + c'
&c.
b' - a' + a"
c' = b' + b"
d' = c' + c"
&c.
b" = a" + a"'
c" = b" + b"'
d" = c" + c!"
&c.
and so on continually. Then eliminating b, b', c, c', &c. from the first
set of equations, we get
b = a + a'
c — a + 2 a' + a"
d = a + 3 a' + 3 a" + a"'
e = a + 4 a' + 6 a" + 4 a'" + a""
f = a + 5 a' + 10 a" + &c.
ic.
Also we have
a' = b — a
a" = c — 2 b + a
of" =d — 3c + 3b — a
&c.
But the letters a', a", a'", &c. are nothing else than A y(,, A?y0, A3y0 . . .
a, b, c . . . being yc, yb y.2 . . . , consequently
y, = y() + A y°
y2 = Jo + 2 A y0 + A?y
y3 = y0 + 3 A y0 + 3 A2yo + ^o
&c.
ANALYTICAL GEOMETRY. Ixxxiii
And
A yo = y, — y0
A2yo = y2 — 2 y, + y0
A3yo = ys-— 3 ya + 3 yj — y0
A'yo = 74 — 4 y3 + 6 ya + 4 y! -{- yc
&c.
Hence, generally, we have
= yo
1 n — 2
These equations, which are of great importance, give the general term
of any series, from knowing its first term and the first term of all the
orders of differences ; and also the first term of the series of nth differ
ences, from knowing all the terms of the series y0, yi, y., . . .
To apply the former to the example in p. (81), we have
A v — 2
jo ~
A3y0 = 6
whence
yx=l — 2x + 2x(x— l)-fx(x— l)(x — 2) = x3 — x2 — 2x-f-l
The equations (A), (B) will be better remembered by observing that
yo ••— \y *) »
provided that in the developements of these powers, we mean by the
exponents of A yfl, the orders of differences, and by those of y the place
in the series.
It has been shown that a, b, c, d . . . may be the values of yx, when
those of x are the progressional numbers
ra, m + h, m + 2 li . . . m + i h
that is
a = ym , b = ym+ h , c - £c.
In the equation (A), we may, therefore, put ym + ih for yx, ym fory0, A ym
f°r A y0> &c. and, finally, the coefficients of the ith power. Make i h = z,
and write A, A2 ... for A ym, A'ym . . . and we shall get
.Vm z = }'m + -A~ + M2^1) ^ • ) Z (Z - h) (Z - 2 h) A3
lxxxiv INTRODUCTION.
This equation will give yx when x = m + z, z being either integer or
fractional. We get from the proposed series the differences of all orders,
and the initial terms represented by A, A2, &c.
But in order to apply this, formula, so that it may be limited, we must
arrive at constant differences ; or, at least, this must be the case if we
would have A, A2 ... decreasing in value so as to form a converging
series : the developement then gives an approximate value of a term cor
responding to ,
x = m + z;
it being understood that the factors of A do not increase so as to destroy
this convergency, a circumstance which prevents z from surpassing a
certain limit.
For example, if the radius of a circle is 1000,
the arc of 60° has a chord 1000,0 _ „. fl
65o 1074,6 A' - I*' A« = - 2,0
70° 1147,2 _23
75° 1217,5
Since the difference is nearly constant from 60° to 75°, to this extent
of the arc we may employ the equation (C); making h = 5, we get for
the quantity to be added to y = 1090, this
}.74,6. z — /sz (z — 5) = 15,12. z — 0,04. z2
So that, by taking z = 1, 2, 3.. . then adding 1000, we shall obtain the
chords of 61°, 62°, 63° ; in the same manner, making z the necessary
fraction, we shall get the chord of any arc whatever, that is intermediate
to those, and to the limits GO0 and 75°. It will be better, however, when
it is necessary thus to employ great numbers for z, to change these limits.
Let us now take
lo<*. 3100 = y = 4913617
m A. = 13987
log. 3110 = 4927604 A * = — 45
13942
log. 3120 = 4941546 _ 45
13897
log. 3130 = 4955443
We shall here consider the decimal part of the logarithm as being an
integer. By making h = 10, we get, for the part to be added to log.
3100, this
1400,95 x z — 0, 225 X z2.
To get the logarithms of 3101, 3102, 3103, &c. we make
z = 1,2, 3....;
and in like manner, if we wish for the log. 3107, 58, we must make
ANALYTICAL GEOMETRY. Ixxxv
z — 7, 58, whence the quantity to be added to the logarithm of 3100 is
10606. Hence
log. 310768 = 5,4924223.
The preceding methods may be usefully employed to abridge the
labour of calculating tables of logarithms, tables of sines, chords, &c.
Another use which we shall now consider, is that of inserting the inter
mediate terms in a given series, of which two distant terms are given.
This is called
INTERPOLATION.
It is completely resolved by the equation (C).
When it happens that A2 = 0, or is very small, the series reduces to
z L yA
TT
whence we learn that the results have a difference which increases propor
tionally to z.
When A 2 is constant, which happens more frequently, by changing z
into z + 1 in (C), and subtracting, we have the genera] value of the first
difference of the new interpolated series ; viz.
First difference A' = £-' + 2 z ~ h + 1 A «
h 2 h2
Second difference A" =: ,---.
If we wish to insert u terms between those of a given series, we must
make
h = n + 1 ;
then making z = 0, we get the initial term of the differences
A.2
A'' —
(11 +
A .
we calculate first A", then A' ; the initial term A' will serve to compose
the series of first differences of the interpolated series, (A" is the constant
difference of it); and then finally the other terms are obtained by simple
additions.
If we wish in the preceding example to find the log. cf 3101,
INTRODUCTION.
3102, 3103 ... we shall interpolate 9 numbers between those which arc
given : whence
u ^ 9
A" = — 0,45
A' = 1400,725.
We first form the arithmetical progression whose first term is A', and
— 0,45 for the constant. The first differences are
1400,725; 1400,725; 1399,375; 1398,925, &c.
Successive additions, beginning with log. 3100, will give the consecutive
logarithms required.
Suppose we have observed a physical phenomenon every twelve hours,
and that the results ascertained by such observations have been
For 0 hours . . . 78 _
12 ... 300 z A2 = 144
24 ... 666
36 ... 1176 510 144.
&c.
If we are desirous of knowing the state corresponding to 4h, 8 h, 12h,
&c., we must interpolate two terms; whence
ti = z, A" = 16, A' = 58
composing the arithmetic progression whose first term is 58, and common
difference 16, we shall have the first differences of the new series, and
then what follow
First differences 58, 74, 90, 106, 122, 138 ...
Series 78, 136, 210, 300, 406, 528, 64G , . .
A Oh, 4h, 8h, 16 h 20 h, 24 ".
The supposition of the second differences being constant, applies almost
to all cases, because we may choose intervals of time which shall favour
such an hypothesis. This method is of great use in astronomy; and
even when observation or calculation gives results whose second differ
ences are irregular, we impute the defect to errors which we correct by
establishing a greater degree of regularity.
Astronomical, and geodesical tables are formed on these principles.
AVe calculate directly different terms, which we take so near that their
first or second differences may be constant ; then we interpolate to obtain
the intermediate numbers.
Thus, when a converging series gives the value of y by aid of that of a
variable x ; instead of calculating y for each known value of x, when the
formula is of frequent use, we determine the results y for the continually
ANALYTICAL GEOMETRY.
increasing values of x, in such a manner that y shall always be nearly of
the same value : we then write in the form of a table every value by the
side of that of x, which we call the argument of this table. For the
numbers x which are intermediate to them, y is given by simple proposi
tions, and by inspection alone we then find the results icqaired.
When the series has two variables, or arguments x and z, the values
ofy are disposed in a table by a sort of double entry ; taking for coordi
nates x and z, the result is thus obtained. For example, having made
z = 1, we range upon the first line all the values ofy corresponding to
x = 1, >', 3...;
we then put upon the second line which z — z gives ; in a third line those
\vhich z = 3 gives, and so on. To obtain the result which corresponds to
x = 3, z = 5
we stop at the case which, in the third column, occupies the fifth place.
The intermediate values are found analogously to what has been already
shown.
So far we have supposed x to increase continually by the same differ
ence. If this is not the case and we know the results
y = a, b, c, d . . .
which are due to any suppositions
X = a
we may either use the theory which makes a parabolic curve pass through
a series of given points, or we may adopt the following:
By means of the known corresponding values
a, a ; b |8 ; &c.
we form the consecutive functions
b — a
c-£fa
y— fl
d — c
6— -,
B = A'~ A
7 — a
B = AlZZ_A
A -
A. =
Ikcj
/4
bcxxviii INTRODUCTION.
r1 -- BI — B
&c.
C'
v — a
and so on.
By elimination we easily get
b = a + A ((3 — a)
c = a + A (7 — a) + B(7— a) (7 — /3)
d = a + A (3— «) + B(3— a) (3—/3) + C(S — a)(3 — /3) (3 — 7)
&c.
and generally
yx = a+A(x — a) + B(x — a)(x_/3) + C (x — a) (x— /3) (x— 7) + &c.
We must seek therefore the first differences amongst the results
a, b, c . . .
and divide by the differences of
a, ft 7 ...
which will give •
A, A19 A2, &c.
proceeding in the same manner with these numbers, we get
B, Bj, Ba, &c.
which in like manner give
C, C,, Cs, &c.
and, finally substituting, we get the general term required.
By actually multiplying, the expression assumes the form
a + a'x + a'x2^-...
of every rational and integer polynomial, which is the same as when we
neglect the superior differences.
The chord of 60° = rad.=rlOOO
=1035
65°. 10' =
A =15
Aj = 14,82
A2= 14,61
B =—0,035
B1 = — 0,031
69°. 0' =1133
We have
a = 0, /3 = 21, 7 = 5^, 8 = 9.
We may neglect the third differences and put
yx = 100 + 15,082 x — 0,035 x2.
Considering every function of x, yx, as being the general term of the
series which gives
x = m, m + b, m + 2 h, &c.
ANALYTICAL GEOMETRY. Ixxxix
if we take the differences of these results, to obtain a new series, the
general term will be what is called the first difference of the proposed
function yx which Is represented by A yx. Thus we obtain this difference
by changing x into x + h in yx and taking yx from the result ; the re
mainder will give the series of first differences by making
x = m, m + h, m -f 2 h, &c.
Thus if '
(x
J x " a + x + h a + x'
It will remain to reduce this expression, or to develope it according to
the increasing powers of h.
Taylor's theorem gives generally (vol. I.)
d y d 2 y h 2
To obtain the second difference we must operate upon A vx as upon <(he
proposed yx, and so on for the third, fourth, &c. differences.
INTEGRATION OF FINITE DIFFERENCES.
Integration here means the method of finding the quantity whose dif
ference is the proposed quantity ; that is to say the general term yx of a
Jin? ym + h> ym + 2h) &c-
from knowing that of the series of a difference of any known order. This
operation is indicated by the symbol 2.
For example
2 (3x2 + x — 2)
ought to indicate that here
h = 1.
A function yx generates a series by making
x = 0, 1, 2, 3 ...
the first differences which here ensue, form another series of which
3 x2 + x — 2
is the general term, and it is
— 2, 2, 12, 28 ...
By integrating we here propose to find yx such, that putting x -f- 1 for
x, and subtracting, the remainder shall be
3 x " + x — 2.
xc INTRODUCTION.
It is easy to perceive that, first the symbols 2 and A destroy one another
as do f and d; thus
2 A f x = f x,
Secondly, that
A (a y) = a A y
gives
2 a y — a 2 y.
Thirdly, that as
A (A t — B u) = A A t — n A u
so is
2 (A t — B u) = A 2 t — B 2 u,
t and u being the functions of x.
The problem of determining yx by its first difference does not contain
data sufficient completely to resolve it ; for in order to recompose the
series derived from yx in beginning with
— 2, 2, 12, 28, &c.
we must make the first term
.Vo = a
and by successive additions, we shall find
a, a — 2, a + 2, a + 12, &c.
in v/hich a remains arbitrary.
Kvery integral may be considered as comprised in the equation (A)
p. 83 ; for by taking
x = 0, 1, 2, 3 . . .
in the first difference given in terms of x, we shall form the series of first
differences ; subtracting these successively, we shall have the second dif
ferences ; then in like manner, we shall get the third, and fourth differ
ences. The initial term of these series will be
A yu, A'-y0. . .
and these values substituted in yx will give yx. Thus, in the example
above, which is only that of page (81) when a = 1, we have
A y0 = — 2, A2 y0 = 4, A3 y0 = 6, A * yQ = 0, &c. ;
which give
yx = y0 — 2 x — x 2 + x 3.
Generally, the first term y0 of the equation (A) is an arbitrary constant,
which is to be added to the integral. If the given function is a second
difference, we must by a first integration reascend to the first difference
and thence by another step to yx ; thus we shall have two arbitrary con
stants ; and in fact, the equation (A) still gives yx by finding As, A3, the
ANALYTICAL GEOMETRY. xci
only difference in the matter being that y0 and A y0 are arbitrary. And
so on for the superior orders.
Let us now find 2 x m, the exponent m being integer and positive.
Represent this developement by
2 x m = p x + qxb + rxc -f- &c.
a, b, c, &c. being decreasing exponents, which as well as the coefficients
p, q, &c. must be determined. Take the first difference, by suppressing
2 in the first member, then changing x into x + h in the second member
and subtracting. Limiting ourselves to the two first terms, we get
o o y o
xm = pahx3-1 + £pa(a — I)h2xa-2 + . . . qbh x"-1 + . ..
But in order that the identity may be established the exponents ought
to give
a — ] = m
a — 2 = b — 1
whence
a = m + 1, b rr m.
Moreover the coefficients give
I = p a h, — % p a (a — 1 ) h — q b ;
whence
P = (ni + 1) h ' q = ~ *'
As to the other terms, it is evident, that the exponents are all integer
and positive ; and we may easily perceive that they fail in the alternate
terms. Make therefore
2xm = pxrn + 1 — £xm + axm-T -f ,Sxm-3 -f 7xm~5 + . ..
and determine «, j3, y ... &c.
Take, as»before, the first difference by putting x + h for x, and sub
tracting : and first transferring
o o
Pv m + 1 __ L v m
X — 2 X ,
we find that the first member, by reason of
p h (m +!) = !,
reduces to
_
' 2.3 4- ' 2.5 6 ' 2.7
To abridge the operation, we omit here the alternate terms of the deve
lopement; and we designate by
1, in, A', A'', &c.
the coefficients of the binomial.
Making the same calculations upon
a x1"-1 + /3 xln-3 + &c.
xcii INTRODUCTION.
we shall have, with the same respective powers of x and of h,
(m-l)a+(m-l).!=2.m^« + (m-l}.'^.. m
+ (m-3)0+(m._S).2^p*...!IL=-?,3+..
A O
+ (m — 4)7+..
Comparing them term by term, we easily derive
m
— A"
" ~~ 2.3.4.5'
. A////
7 ~~ 6.6.7
&c.
whence finally we get
+ A""ch5xm-5 + Avi dh7xm-7+...(D)
This developement has for its coefficients those of the binomial, taken
from two to two, multiplied by certain numerical factors a, b, c . . ., which
are called the numbers of Bernoulli, because James Bernoulli first deter
mined them. These factors are of great and frequent use in the theory
of series ; we shall give an easy method of finding them presently. These
are their values
J^
= 12
b = -
120
1
~ 252
240
1
6 ~ 132
691
f =
32780
I
12
h = -
8160
. _ 43867
~ 14364
&C.
ANALYTICAL GEOMETRY.
which it will be worth the trouble fully to commit to memory.
From the above we conclude that to obtain 2 xm, m being any number,
integer and positive, we must besides the two first terms
xm+1 xm
(m + 1) h 2~
also take the developernent of
(x + h) m
reject the odd terms, the first, third, fifth, &c. and multiply the retained
terms respectively by
a, b, c . . .
Now x and h have even exponents only when m is odd and reciprocally :
so that we must reject the last term h m when it falls in a useless situation ;
the number of terms is | m + 2 when m is even, and it is | (m + 3) when
m is odd ; that is to say, it is the same for two consecutive values of m.
Required the integral ofx 10.
Besides
x11
_± ___ 1 ,,10
11 h
we must develope (x + h) % retaining the second, fourth, sixth, &c. terms
and we shall have
10x9ah+ 120x7bh3 + 252x5ch5 + &c.
1 herefore
In the same manner we obtain
2xo _ *
• h
2X1 -
X*
x
2 h
2
v X 5
X3
x 2 h x
3~h
2 + 6 '
2 X3 -
x4
x3 hx2
4h
2 4
2 X4-
x5
x4 hx3 h3x
5 h
4 3 30
V v 5 —
— A. ^_
x6
x5 5hx4 h3x2
6h
2 12 12 '
2 Xfi -
x7
x6 h x5 h3xl- h5x
7 h
2 2 6 42 '
2 X7 =
x8
8li
x 7 7 h x c 7h3x4 h5
" 2 ' 12 -2T~ + "I
xciv INTRODUCTION
- 8 _ __. _ _ , _ _
~ y h " 2 + 3 15 9 " 30"
9-^-^4. 3 h x' _ 7 h 3 x ° ti5x4 __ 3 h7x 2
= 10h~" 2 4 10 2 ' 20
x11
5 x10 = —— - -- &c. as before,
11 h
&c.
We shall now give an easy method of determining the Numler of
Bernoulli a, b, c . . . In the equation (D) make
X = h = 1;
2 x m is the general term of the series whose first difference is x ™. We
shall here consider 2. x° = ], and the corresponding series which is that
of the natural numbers
0, 1, 2, 3 ...
Take zero for the first member and transpose
JL i
m + 1 "
which equals
I — m
Then we et
= a m
+ b A" + c A Iv + cl A * -f . . . + k m.
2 (m+
By making m = 2, the second member is reduced to am, which gives
Making m = 4, we get
3 = 4 a + b K"
10
m — 1 m — 2 .
4 a + m . — . — b
— 4 a + 4 b
= f + 4 b.
Whence
b - _L
120*
Again, makiug m = 6, we get
5
— = 6 a + b A" + c A
= 6a+ 20 b +6c
= i — -i- + o c
ANALYTICAL GEOMETRY. xcv
which gives
s\ •<«« • , _ .
" 252'
and proceeding thus by making
m = 2, 4, 6, 8, &c.
we obtain at each step a new equation which has one term more than the
preceding one, which last terms, viz.
2 a, 4 b, 6 c, . . . m k
will hence successively be found, and consequently,
a, b, c . . . k.
Take the difference of the product
yx = (x — h)x (x + h) (x + 2h)...(x+ih),
by x + h for x and subtracting ; it gives
A yx = x (x + h) (x + 2 h) ... (x + i h) x (i + 2) h;
dividing by the last constant factor, integrating, and substituting for yx
its value, we get
2x (x + h) (x+ 2h)...(x + ih)
Xx'(x
This equation gives the integral of a product of factors in arithmetic
progression.
Taking the difference of the second member, we veiify the equation
v _ 1 _ = __ -» _ ;
x (x + h) (x + 2 h) . . .(x + i h) i h x (x + h) . . . [x + (i — 1) h}
which gives the integral of any inverse product
Required the integral of a.*.
Let
v - n X
}x — » •
Then
Ay* = ax (ah — 1)
whence
yx = 2ax (u" — I) = ax;
consequently
ax
5 a x = —r - - + constant.
ah — 1
Required the integrals qfs'm. x, cos. x.
Since
cos. B — cos. A = 2 sin. % (A + B). sin. £ (A — B)
A cos. x = cos. (x + h) — cos. x
hx h
= — 2 sin. (x + -'-) sin.
xcvi INTRODUCTION.
Integrating and changing x + -— - into z, we have
m
(— *)
2 sm. z = — cos. -f- constant.
• h
In the same way we find
2 cos. z = { f- constant.
h
2 sin.-
When we wish to integrate the powers of sines and cosines, we trans
form them into sines and cosines of multiple arcs, and we get terms of
the form
A sin. q x, A cos. q x~
Making
q x rr x
the integration is performed as above.
lieguired the integral of a product, viz.
Assume
2(uz) = u2z + t
u, z and t being all functions of x, t being the only unknown one. By
changing x into x + h in
u 2 z + t
u becomes u + A u, z becomes z + A z, &c. and we have
u2z+uz + Au2(z + Az) + t+At;
substituting from this the second member
u 2 z + t,
we obtain the difference, or u z ; whence results the equation
0 = Au2(z + AZ) + At
which gives
t = — 2 £A u 2 (z + A z)}.
• Therefore
2 (u z) = U 2 Z — 2 {A u . 2 (Z + A z)]
which is analogous to integrating by parts in differential functions.
There are but few functions of which we can find the finite integral ;
when we cannot integrate them exactly, we must have recourse to series.
Taylor's theorem gives us
dy. , d2y hs
A yx = , h + -r-v -^- + &c.
J* dx dx2 ^
ANALYTICAL GEOMETRY.
by supposition. Hence
yx = h 2 y' + ~ 2 y" + &c.
Considering y' as a given function of x, viz. z, we have
y ' = *
y/// __ 7n
&c.
and
yx = /y'dx = /zdx
whence
h2
/z d x = h 2 z + — 2 z' + &c.
2f
which gives
2 z = h-1 /"z d x — 4 2 z' — 7- h 2 2 z" — &c-
o
This equation gives 2 z, when we know z', 2 z'', &c. Take the dif
ferentials of the two numbers. That of the first 2 z will give, when di
vided by d x, 2 z'. Hence we get 2 z", then 2 z"', &c. ; and even without
making the calculations, it is easy to see, that the result of the substitution
of these values, will be of the form
2 z = h-'/z d x + A z + B h z' + C h 2 z" + &c.
It remains to determine the factors A, B, C, &c. But if
z = xm
we get
/z d x, z', z", &c.
and substituting, we obtain a series which should be identical with the
equation (D), and consequently defective of the powers m — 2, m — 4,
so that we shall have
_/-zdx z a h z' b h' z'" , cW"" dhV""" ,
h 2~H ~T~ ~TT 2.3.4 2... 6
a, b, c, &c. being the numbers of Bernoulli.
For example, if
z = 1 x
yix.dx = x 1 x — x
z' = x-1
z" = &c.
xcviii INTRODUCTION
consequently
2lx = C-fxlx — x — £lx + a x-1 + b x~3 + c x-/ + Sec.
The series
a, b, c . . . k, 1,
having for first differences
i' h' c' I'
B 9 V 9 v •••K
we have
b = a + a'
c = b + b
(1 = c + c'
ate.
i = k + k'
equations whose sum is
1 = a + a' + b' + c' + . . . k'.
If the numbers a', b', c', &c. are known, we may consider them as being
the first differences of another series a, b, c, &c. since it is easy to com
pose the latter by means of the first, and the first term a. By definition
we know that any term whatever 1', taken in the given series a', b', c', &c.
is nothing else than A 1, for 1' = m — 1 ; integrating
T = A 1
we have
21' =2 1
or
2 1' = a' + b' + c' . . . + k',
supposing the initial a is comprised in the constant due to the integra
tion. Consequently
The integral of any term whatever of a series^ we obtain the sum of all
the terms that precede it, and have
2 yx = y0 + yi + y« + • • • y * - 1.
In order to get the sum of a series, we must add yx to the integral ; or
which is the same, in it must change x into x + 1, before we integrate.
The arbitrary constant is determined by finding the value of the sum y0
when
x = 1.
We know therefore how to Jind the summing term of every series whose
general term is known in a rational and integer function ofx.
Let
yx = A x m — B x n + C
m ;and n being positive and integer, and we have
A2xra — B 2 x" + C 2 x°
ANALYTICAL GEOMETRY. xcix
for the sum of the terms as far as yx exclusively. This integral beino
once found by equation D, we shall change x into x + 1, and determine
the constant agreeably.
For example, let
y«= x(2x-l);
changing x into Z + 1, and integrating the result, we shall find
A -.-• 3 I O 9
2 2x3 + 3 2x+ 2X° = —
= x .
2.3
x + 1 4 x — ]
2 3
there being no constant, because when x = 0, the sum = 0.
The series
lm, 2m, 3m...
of the mth powers of the natural numbers is found by takimr 2 xm (equa
tion D); but we must add afterwards the xth term which is xm; that is to
say, it is sufficient to change — £ xm, the second term of the equation
(D), into £ xm; it then remains to determine the constant from the term
we commence from.
For example, to find
S = 1 + 2* + 32 + 4» + .,.x'
we find 2 x 2, changing the sign of the second term, and we have
x3 x2 x x+ 1 2x + 1
S--3 +-2 + 6- = X'-3-- — iH
the constant is 0, because the sum is 0 when x = 0. But if we wish to
find the sum
S' = (n + I)2 + (n + 2)2 + ...x*
S' = 0, whence x = n — 1, and the constant is
n — 1 2 n — 1
2 ~3 >
which of course must be added to the former ; thus giving
S' = (n + 1)* + (n + 2)' +...x*
x + 1 2x -f 1 n — 1 2 n — 1
3 2 ~2~' -- 3~
= -0- X {x.(x+ 1). (2x + l)_n.(n— 1) (2 n — I)
This theory applies to the summation ofjgurate numbers, of the dif
ferent orders : —
c INTRODUCTION.
First order, 1.1.1.1.1. 1 . 1 , &c.
Second order, 1.2.3.4.5. 6 . 7 , &c.
Third order, 1.3. 6 . 10 . 15 . 21 . 28 , &c.
Fourth order, 1 . 4 . 10 . 20 . 35 . 56 . 84 , &c.
Fifth order, 1 . 5 . 15 . 35 . 70 . 126 . 210, &c.
and so on.
The law which every term follows being the sum of the one immediate
y over it added to the preceding one. The general terms are
First, 1
Second, x
Third, X'(X2+1)
r . x (x + 1) (x + 2)
Fourth, • v oV
Dtn x.(x+ 1) (x + 2)...x + p — 2
1 . 2 . 3 . . . p — 1
To sum the Pyramidal numbers, we nave
S = 1 + 4 + 10 + 20 + &c.
Now the general or xth term in this is
yx = -1 . x (x + 1) (x + 2).
But we find for the (x — 1)* term of numbers of the next order
2l (x — !) x (x + 1) (x + 2) ;
finally changing x into x + 1, we have for the required form
S = ^-x.(x + l)(x + 2)(x +3).
Since S = 1, when x = 1, we have
1 = 1 + constant, consequently
.\ constant = 0.
Hence it appears that the sum of x terms of the fourth order, is the
xtb term or general term of the fifth order, and vice versa ; and in like
manner, it may be shown that the xth term of the (n + l)th order is the
sum of x terms of the nth order.
Inverse Jigurate numbers are fractions which have 1 for the numerator,
and a figurate series for the denominator. Hence the xth term of the pth
order is
1 . 2 . 3 . . . (p — _!)_
x (x + 1).. .x + p — 2
ANALYTICAL GEOMETRY. oi
and the integral of this is
(p — 2)x(x +l)...(x + p — 3)'
Changing x into x+1, then determining the constant by making
x = 0, which gives the sum = 0, we shall have
p— 1.
and the sum of the x first terms of this general series is
p— 1 1.2.3...(p — 1)
p _ 2 (p — 2) (x + 1) (x + 2) . . . (x + p — 2)*
In this formula make
p - 3, 4, 5 ...
and we shall get
1 4. * 1 ! 4. 1 4. 1-2 - 2 2
1 ' ~ 3 " ~ 6 "r 10 "* ' x (x + 1) 1 x+1
_!_ I !_ 1 1.2.3 _ 3 3
1 " 4 * 10 + 20 + " ' x (x + J ) (x + 2) " 2 (x + 1) (x + 2)
1 1 , J_ _L 1 .2.3.4 _ 4 2.4
T+ 5 +T0 "*" 35 + *"x(x+l)(x+2) (x + 3) ~ 3" (x+ J)...(x + 3)
1 1^ 1 ± 1.2.3.4.5 5 2.3.5
T+ 6 + 21 + 56 + *"x(x+l)...(x + 4) ~ 4 (x+1) . ..(x + 4)
and so on. To obtain the whole sum of these series continued to infinity,
we must make
X = CD
which gives for the sum required the general value
P-l
P-2
which in the above particular cases, becomes
2345
1' 2' 3' 4' &C>
To sum the series
sin. a + sin. (a + h) + sm. (a + 2 h) + . . . sin. (a + x — 1 h)
we have
cos. (a + h x -- -J
? sin. (a + x h) = C -- j - •
2sin.J
changing x into x + 1, and determining C by the condition that x = — J
makes the sum = zero, we find for the summing-term.
cos.
. (a — ^-) — cos. (a + h x +
cii INTRODUCTION.
or
sin.
. / , h x . h (x + 1)
sin. (a + — xj sin. — ^ '•
In a similar manner, if we wish to sum the series
cos. a + cos. (a + h) -f- cos. (a -f 2 h) + . . . cos. (a + x — 1 h}
we easily find the summing-term to be
sin. (a. ^\ — sin. ( a + h x -f -- \
\ '- d / \ lil '
2 sin. A
or
h ^ h (x 4. 1)
__ .
nn. ¥
cos. t - , g
A COMMENTARY
ON
NEWTON'S PRINCIPIA,
SUPPLEMENT
TO
SECTION XL
460 PROP. LVII, depends upon Cor. 4 to the Laws of Motion,
which is
If any number of bodies mutually attract each other, their center of gra
vity will either remain at rest or will move uniformly in a straight line.
First let us prove this for two bodies.
Let them be referred to a fixed point by the rectangular coordinates
*> y ; x', y',
and let their masses be
(*» /*'.
Also let their distance be ?, and f (§) denote the law according to which
they attract each other.
Then
will be their respective actions, and resolving these parallel to the axes of
abscissas and ordinates, we have (46)
VOL. II.
A COMMENTARY ON [SECT. XL
Hence multiplying equations (1) by ^ and those marked (2) by ft,' and
adding, &c. we get
dt
= 0,
and
dt2
and integrating
d x , d x'
'dt dt •
Now if the coordinates of the center of gravity be denoted by
x, y,
we have by Statics
- /a x jf x7
_
+
+ ft
d x _ 1 / d x , dx\
' d t /i + fjf ' \ d t d t /
and
^ y
dt" =.
But
d x d y
dl' "dT
represent the velocity of the center of gravity resolved parallel to the axes
of coordinates, and these resolved parts have been shown to be constant
Hence it easily appears by composition of motion, that the actual velocity
of the center of gravity is uniform, and also that it moves in a straight
line, viz. in that produced which is the diagonal of the rectangular par
allelogram whose two sides are d x, d y.
If
c = 0, c' = 0
then the center of gravity remains quiescent.
BOOK!.] NEWTON'S PRINCIPIA. 3
461 The general proposition is similarly demonstrated, thus.
Let the bodies whose masses
I*', p", /", &c.
be referred to three rectangular axes, issuing from a fixed point by the
coordinates
x' v' 7'
A ? y 5 z
/
y
&c.
x'", y"', -z!"
Also let
^ij 2 be the distance of //, //'
&C. &C.
and suppose the law of attraction to be denoted by
f-(*i,2)» f(fi,s)> f (fo,s)» &c-
Now resolvin the attractions or forces
&c.
parallel to the axes, and collecting the parts we get
d2x' x' — x" , ,,ff, .x'-
= /* I (?i o ) • -4- U> I ( ?i -j)
&C.
&c-
t O, O 0.1 Q
jl, J j •, 9
&c. = &c.
Hence multiplying the first of the above equations by [jft the second by
a7', and so on, and adding, we get
gM^ + ^d'x" + it!" d g x'" + &c. _ .
~dT2~
Again, since it is a matter of perfect indifference whether we collect the
forces parallel to the other axes or this ; or since all the circumstances are
similar with regard to these independent axes, the results arising from
similar operations must be similar, and we therefore have also
fif d2 y' + tt," d2 y" + f*'" d 8yw + &c. _
dt8 °'
d 2 z" + »'" d g T!" + &c. _
dt« ~~°-
A 2
4 A COMMENTARY ON [SECT. XI.
Hence by integration
, dx' , ,.dx" , ^dx'" ,
""d7 + /A -dT+/i "dT + &c' = c
d v' d v" d v"'
'/-^T+*"TT+'/""dT + &c=<;'
1 rl •/' rl •/' <1 7///(
. U Z .. U t, ... U. £1 0 ..
*'dl +fJ> -dT + " -dF+&c'=c'
But x, y, z denoting the coordinates of the center of gravity, by statics
we have
- - / *' + V>" *" + V-'" *'" + &c.
tf + p» + p"'' + &C.
_ p'f + ^ y/r + ^w y/7/ + &c.
tf + X' + <""' + &c.
_ _ p' z' + ^ z^ + (,/» i>" + &c.
p' + p," + ^' + &c.
and hence by taking the differentials, &c. we get
dx c
d t " p! + fjf' + v!" + &c.
d y (^
^"
d t /*' + n" + u!" + &c.
that is, the velocity of the center of gravity resolved parallel to any three
rectangular axes is constant. Hence by composition of motion the actual
velocity of the center of gravity is constant and uniform, and it easily ap
pears also that its path is a straight line, scil. the diagonal of the rectan
gular pai'allelopiped whose sides are d x, d y, d z.
462. We will now give another demonstration of Prop. LXI. or that
Of two bodies the motion of each about the center of gravity, is the same
as if that center was the center of force, and, the law of force the same as
that of their mutual attractions.
Supposing the coordinates of the two bodies referred to the center of
gravity to be
we have
x = x + x, ^ x' = x + x,,
y=~y + y/j y' = y + y/,
Hence since
d x d y
eft ' dT
BOOK I.] NEWTON'S PRINCIPIA.
are constant as it has been shown, and therefore
— 0
"
d2x dy
£. —
, _ . _ .
dt2 ' ' cit«
we have
<Px _dgx/
dt2 "TF
cPy _ d*y/
dt2 " dt«"
and we therefore get (46)
But by the property of the center of gravity
being the distance of /*' from the center of gravity. We also have
f T"
Hence by substitution the equations become
Similarly we should find
and
Hence if the force represented by
were placed in the center of gravity, it would cause /"•' to move about it as
a fixed point; and if
were there residing, it would cause ^ to centripetate in h"ke manner.
Moreover if
A 3
6 A COMMENTARY ON [SECT. XI
then these forces vary as
a/n, an;
so that the law of force &c. &c.
ANOTHER PROOF OF PROP. LXII.
463. Let p, [i! denote the two bodies. Then since & has no motion
round G (G being the center of gravity), it will descend in a straight
line to G. In like manner p' will fall to G in a straight line.
Also since the accelerating forces on p, tf are inversely as /*, p' or
directly as G A», G //, the velocities will follow the same law and corre
sponding portions of G ^ G tf will be described in the same times ; that
is, the whole will be described in the same time. Moreover after they
meet at G, the bodies will go on together with the same constant velocity
with which G moved before they met.
Since here
a. will move towards G as if a force
^^
or
Hence by the usual methods it will be found that if a be the distance
at which <«. begins to fall, the time to G is
0 + pf) a 2 v
^l '2V2
and if a' be the original distance of/*', the time is
(ft + X) of * cr
„* '2V2'
But
a : a' : : p' : p
therefore these times are equal, which has just been otherwise shown.
BOOK I.] NEWTON'S PRINCIPIA. 7
ANOTHER PROOF OF PROP. LXII1.
464. We know from (461) that the center of gravity moves uniformly
in a straight line; and that (Prop. LVII,) p and fjf will describe about G
similar figures, p moving as though actuated by the force
and Q as if by
Hence the curves described will be similar ellipses, with the center of
force G in the focus. Also if we knew the original velocities of p and y!
about G, the ellipse would easily be determined.
The velocities of /* and [jf at any time are composed of two velocities,
viz. the progressive one of the center of gravity and that of each round G.
Hence having given the "whole original velocities required to find the separate
•parts of them,
is a problem which we will now resolve.
Let
V, V
be the original velocities of /a, //, and suppose their directions to make
with the straight line p yf the angles
a, of.
Also let the velocity of the center of gravity be
v
and the direction of its motion to make with p fjJ the angle
a.
Moreover let
v, v'
be the velocities of /*, /// around G and the common inclination of their
directions to be
6.
Now V resolved parallel to p // is
V cos. «.
But since it is composed of v and of v it will also be
v cos. a -f- v cos. &
.'. V COS. a = v COS. a -}- v COS. &.
In like manner we get
V sin. « = v sin. a + v sin. 6.
A 4
8 A COMMENTARY ON [SECT. XL
and also
V' cos. a! — v cos. a — v' cos, 6
V sin. a! =r v sin. a. — v' sin. 6.
Hence multiplying by /*, At', adding and putting
At v = A1' v'
we get
At V COS. a + At' V COS. a' = (A& + //) V COS. a
and
At V sin. a + At' V sin. «'=(/«, + ///)
Squaring these and adding them, we get
At2 V2 -f- y^V'2 + 2A*At' VV'cos.(a-— a') =
winch gives
v=
(if) V COS. a A
At') v sin. a J
At + X
By division we also have
— _ ft V sin. a -f- (*' V' sin. a'
tan. u — fj - /r^pr/ / •
^ V cos. a + // V cos. a
Again, from the first four equations by subtraction we also have
V cos. a — V7 cos. a' — (v + v') cos. 6 = v . ^ , ^ cos. 0
^
V sin. a — V7 sin. a' r= (v + V) sin. 0 = v . ^ , A sin. ^
p
and adding the squares of these
V* + V/2 — 2 V V'cos. (a — aO= v2
whence
v = 7. ViV2 + V/2 — 2 VV'cos. (a— «')
f* + «'
+ V/2 — SVV'COS. (a — a')
and by division
V sin. a — V sin. a'
tan. 0 = ^ - v^ - , .
V COS. a — V7 COS. a'
Whence are known the velocity and direction of projection of /* about
G and (by Sect. III. or Com.) the conic section can therefore be found ;
and combining the motion in this orbit with that of the center of gravity,
which is given above, we have also that of/*.
465. Hence since the orbit of fj> round (*' is similar to the orbit of
tt round G, if A be the semi-axis of the ellipse which /* describes round
BOOK I.] NEWTON'S PRINCIPIA. 9
G, and a that of the ellipse which it describes relatively to /*' which is also
in motion j we shall have
A : a : : IM/ : /* + /&'.
466. Hence also since an ellipse whose semi-axis is A, is described by
the force
we shall have (309) the periodic time, viz.
T - 2 A'^ff __ 2 g A s (,a + Ap
2 ff
V (A* + it,1) '
467. Hence we easily get Prop. LIX.
For if At were to revolve round /*' at rest, its semi-axis would be a, and
periodic time
.-. T : T' : : V / : V (^ + t*').
468. PROP. LX is also hence deducible. For if /* revolve round (if a*
rest, in an ellipse whose semi-axis is a', we have
and equating this with T in order to give it the same time about /«•' at rest
as about &' in motion, we have
.-. a : a' : : (/& + ^') : ^' .
ANOTHER PROOF OF PROP. LXIV.
469. Required the motions of the bodies whose masses are
ft, //, p», p."', &C.
and which mutually attract each other with forces varying directly as the
distance.
Let the distance of any two of them as p, ,«/> be j ; then the force of (i!
on <j. is
10 A COMMENTARY ON [SECT. XI.
and the part resolved parallel to x is
/ ^ L~ X . , «»
(L1 s • —— = f' (X — X').
In like manner the force of [*" on p, resolved parallel to x, is
p" (x — x")
and so on for the rest of the bodies and for their respective forces resolved
parallel to the other axes of coordinates.
Hence
^ = o! (x — x') + ?,» (x - x") + &c.
T=fi (*' — x) + /' (x' — x") + &c.
~ = ft (x" - x) + nf (x" — x') + &c.
&c. = &c.
which give
^~ = (p, + f*' + p» + &c.) x— (it, x + ^ x' + &c.)
/X' + &C.)
= (^ + ^' + X' + &C.) X"— (^ X + ft,' X' + &C.)
&c. = &c.
Or since
(J, X + (*' X7 + &C. = ([* + iff + &C.) X
making the coordinates of the center of gravity
x> y, z",
we have
&c. = &c.
In like manner, we easily get
?= (" + <*'+&c.)(y — y)
BOOK I.] NEWTON'S PRINCIPI A. 11
&c. = &c.
and also
f = (,* + v' + &c.) (z — ~z)
&c. = &c.
Again,
x — x , y — y , z — z
x'— x~,y' — y, z' — z
&c. &c. &c.
are the coordinates of /tt, /»', /"/', &c. when measured from the center of
gravity, and it has been shown already that
d 2 (x — x) _ d^x
~~d t2 ~ dT2
d2(y-y) _ d^y
dt2 "dts
d2(z — z) _ d2z
dt« ~dt2
and so on for the other bodies. Hence then it appears, that the motions
of the bodies about the center of gravity, are the same as if there were but
one force, scil.
(//, + (i! + &c.) X distance
and as if this force were placed in the center of gravity.
Hence the bodies will all describe ellipses about the center of gravity,
as a center; and their periodic times will all be the same. But their
magnitudes, excentricities, the positions of the planes of their orbits, and
of the major axes, may be of all varieties.
Moreover the motion of any one body relative to any other, will be
governed by the same laws as the motion of a body relative to a center
of force, which force varies directly as the distance ; for if we take the
equations
- = (0, + ^ + &c.) (x — x)
12 A COMMENTARY ON [SECT. XL
and subtract them we get.
f|2 (X v>^
•(dt2 ^ = (A* + ft + &C.) (X - X')
and similarly
d2(v v')
ch^ = ^ + /v + &c-} (-v - y')
and
^s/7 _/\
^t. = (*» + ^ + &C.) (Z — 2').
Hence by composition and the general expression for force (yr|) ^
readily appears that the motion of & about p'y is such as was asserted.
470. Thus far relates merely to the motions of two bodies ; and these
can be accurately determined. But the operations of Nature are on a
grander scale, and she presents us with Systems composed of Three, and
even more bodies, mutually attracting each other. In these cases the
equations of motion cannot be integrated by any methods hitherto dis
covered, and we must therefore have recourse to methods of approxi
mation.
In this portion of our labours we shall endeavour to lay before the
reader such an exposition of the Lunar, Planetary and Cometary Theories,
as may afford him, a complete succedaneum to the discoveries of our
author.
471. Since relative motions are such only as can be observed, we refer
the motions of the Planets and Comets, to the center of the sun, and the
motions of the Satellites to the center of their planets. Thus to compare
theory with observations,
// is required to determine the relative motion of a system of bodies, about
a body considered as the center of their motions.
Let M be this last body, /*, (*•', /«,", &c. being the other bodies of which
is required the relative motion about M. Also let
C, n,7
be the rectangular coordinates of M ;
£+ x, n + y, 7 + z;
£+x'n + y',7+z';
&c.
those of ft, /«/, &c. Then it is evident that
x, y, z ;
Tff v/ y'
x > y > z
&c.
BOOK I.] NEWTON'S PRINCIPIA. 13
will be the coordinates off*, &', &c. referred to M.
Call ft /, &c.
the distances of p, ///, &c. from M; then we have
• f = v (x ' * + y - 4- z i)
ft /> &c. being the diagonals of rectangular parallelepipeds, whose sides
are
x, y, z
Vf I/ 1>
x j y > z
&c.
Now the actions of /w, /»', ^", &c. upon M are
At (jf [jf
~~Z ) t 2 J "V7~2 ) O^^*
and these resolved parallel to the axis of x, are
V* X ft' x.' /A" xv
7F'Tr'77ir'&c'
Therefore to determine £, we have
dT2^ = 73 + "73" + T7^ + &c.
the symbol 2 denoting the sum of such expressions.
In like manner to determine n, 7 we have
dt2 "7^'
Q y $& z
dl2111 2*7T>
The action of M upon /*, resolved parallel to the axis of x, and in the
contrary direction, is
_Mx
Also the actions of ^', A*", &c. upon ^ resolved parallel to the axis of x
are, in like manner,
tf (*' — x) ^ (*" — . x) ^" (X/// _ x)
fd.m generally denoting the distance between ///" ••••" and ///" ••• «
But
x — y)2+ (2' _ z)
to» = V (X" — X)2+ (y"__
&C. = &C.
A COMMENTARY ON [SECT. XL
f 1§8 = V (X" — x') 2 + (y" — y') 2 + (z" —
and so on.
Hence if we assume
^ _ p.p.' i^y." &c
£0,1 go,2
i, 2 fl,3
§2,3
&C.
and taking the Partial Difference upon the supposition that x is the only
variable, we have
J- . (**\ = "x(x'-») + £(*> -x) &c>
the parenthesis ( ) denoting the Partial Difference. Hence the sum of
all the actions of /»', /»", &c. on /i is
JL.fJil
A* Vdx/'
Hence then the whole action upon /A parallel to x is
d.2 (I + x) _ J_ xd_Xv MX
d t2 = /t6 ' Vdx/ ~ f3 ;
But
d2x 1 /d >.\ MX ^x .
" ~~" '^
Similarly, we have
d t2 " n Vdy^ ^3 ' gr
1 /d Xx M z ^ z
~^l 2-TF (3)
t
If we change successively in the equations (1), (2), (3) the quantities
,«•, x, y, z into
(*', x', y', z';
..// v// v// „// .
f' 5 x j y 5 z »
&c.
and reciprocally ; we shall have all the equations of motion of the bodies
^ /a", &c. round M.
BOOK I.] NEWTON'S PRINCIP1A. 15
If we multiply the equations involving £ by M + 2. p ; that in x, by
u. ; that in x', by /a', and so on ; and add them together, we shall have
. d2? /d x\ /dx\ /dXx d2x
(M + s.^jpM (dx) + (d~,
But since
d Xx ,» g' (X- - X)
/ x = ,» g X - X &^
\dx/ J0
. &c
and so on in pairs, it will easily appear that
x d2^ d2 x
•••(M + 2-^dT' = -2^dT^
whence by integrating we get
d ? - c ^ d t - --Jl
; ~ M + 2.^ M +
and again integrating
, 2. fj. x
= a + ~
a and b being arbitrary constants.
Similarly, it is found that
These three equations, therefore, give the absolute motion of M in
space, when the relative motions around it of p, /, /a.", &c. are known.
Again, if we multiply the equations in x and y by
and
2 . ^ X-
in like manner the equations in x' and y' by
16 A COMMENTARY ON [SECT. XL
and
. 2. fjj X
and so on.
And if we add all these results together, observing that from the nature
of X, (which is easily shown)
and that (as we already know)
/d X\ /d X\
2. (-y-) = 0, 2. (--) = 0,
\dx/ \d y/
we have
xd2y — y d2x 2.,/*x d2y
- -- - -
y ' d2 x
' '" (J"
M + 2,<T *dt2
and integrating, since
/(xd2y — yd2x) =/xd2y— /yd2x
= x d y — yd xdy — (ydx — yd xdy)
— xdy — ydx,
we have
x d v — v d x 2,/u-x dy
2 i^ — const, -f- *r? • 2 . (& . •*•*
dt M + 2 . ,«• dt
2. [*> y d x
~ M + 2. A& ' * 5Tt
Hence
,., xdy — ydx xdy — ydx dx
C = M . 2 . IL . y . * U2 . A* X 2 u, . ^— 5-^ 1- 2 . A* y X 2 . /* -,-
— 2 . ^ x X 2 .
c being an arbitrary constant.
In the same manner we arrive at these two integrals,
(yy-y) (d
7 and c" being two other arbitrary constants.
BOOK I.] NEWTON'S PRINCIPIA. 17
Again, it' we multiply the equation in x by
i
2 p d x — 2
2. fjt, d x
'**:-- -
M + 2.^'
the equation in y by
-.
M + 2.^'
the equation in z by
o j « 2 . «, d Z
2 /* d z — 2 p . .—- - •
M + 2 . (J, '
it' in like manner we multiply the equations in x', /, z' by
• M + 2.^
2t 1 / r* t •" • A^ Cl V
/// d y' — 2 u, . -_l-:_z *-•
M + 2. (j.
£•»/!/ ^v/ P « /* d Z
2 &' d z' — 2 ,<// . =i-= :
M + 2. (i,9
respectively, and so on tor the rest ; and add the several results, observ
ing that
we get
2 v
_
dt2 = M + 2^" ' dt
. 2 s . ^ d y ^ ^d 2 y 2 2 . ^ d z ^ >d2z
' 2 h ' "
and integrating, we have
2 » P T-T5 = const. + - — — — p-jt-— I
d t2 (M -f s/tt) d t
+ 2 M 2— + 2 X,
which gives
, «, dx'+dy^dz* , . ( (dxr-dx)2+(d>
-5Tr- -+2.^.|V- ^_, , j
— f 2 M 2. -^- + 2x| (M + 2 /(*)
i f J
h being an arbitrary constant.
VOL. IT. B
18 A COMMENTARY ON [SECT. XI.
These integrals being the only ones attainable by the present state of
analysis, we are obliged to have recourse to Methods of Approximation,
and for this object to take advantage of the facilities afforded us by the
constitution of the system of the World. One of the principal of these
is due to the fact, that the Solar System is composed of Partial Systems,
formed by the Planets and their Satellites : which systems are such, that
the distances of the Satellites from their Planet, are small in comparison
with the distance of the Planet from the Sun : whence it results, that the
action of the Sun being nearly the same upon the Planet as upon its Satel
lites, these latter move nearly the same as if they obeyed no other action
than that of the Planet. Hence we have this remarkable property,
namely,
472. The motion of the Center of Gravity of a Planet and its Satellites,
is very nearly the same as if all the bodies formed one in that Center.
Let the mutual distances of the bodies ^, &', p", &c. be very small
compared with that of their center of gravity from the body M. Let
also
X = x + X, ; y = y -f- y, ; z = z + z,.
x' = x" + x/ ; y' = y + y/; z' = "z + z/;
&c.
x, y, z being the coordinates of the center of gravity of the system of
bodies p, (if9 ,»", &c. ; the origin of these and of the coordinates x, y, z ;
x', y', z', &c. being at the center of M. It is evident that x,, y/5 z, ;
x/, y/, z/, &c. are the coordinates of (i>, pf, &c. relatively to their center of
gravity ; we will suppose these, compared with x, y, z, as small quanti
ties of the first order. This being done, we shall have, as we know by
Mechanics, the force which sollicits the center of gravity of the system paral
lel to any straight line, by taking the sum of the forces which act upon the
bodies parallel to the given straight line, multiplied respectively by their
masses, and by dividing this sum by the sum of the masses. We also
know (by Mech.) that the mutual action of the bodies upon one another,
does not alter the motion of the center of gravity of the system ; nor does
their mutual attraction. It is sufficient, therefore, in estimating the forces
which animate the center of gravity of a system, merely to regard the
action of the body M which forms no part of the system.
The action of M upon //., resolved parallel to the axis of x is
MX
BOOK I.] NEWTON'S PRINCIPIA. 19
the whole force which sollicits the center of gravity parallel to this straight
line is, therefore,
Substituting for x and g their values
x It + x,
Ux + x/) 2 + (y + y/)H- (z + Z/)2P
If we neglect small quantities of the second order, scil. the squares and
products of
"V V 7 • Y ' V ' <7 ' • %7f*
/' y /' / ' / 5 y/ j "i > otc.
and put
7 = V (x 2 + P + z"2)
the distance of the center of gravity from M, we have
- = * 4- 3- 3 x (x x, + "y y/ + z z,)
e3 f3 ~s3 73
for omitting x 2, y 2 &c., we have
p = (i + X/) X K?)2 + 2 (x x, + y y/ + z Z/)} ~f nearly
= (x+xy) X J(7) ~3 — 3 (7) ~ 5 (x x/ + y y/ + z~zj nearly
x + x/ 3 x - -
= — "^\T~ "" — • (x x/ + y y/ "i z z/) nearly.
Again, marking successively the letters x/5 y/s z/5 with one, two, three,
&c. dashes or accents, we shall have the values of
But from the nature of the center of gravity
,
— __ _i nearJv
3
Thus the center of gravity of the system is sollicited parallel to the
axis of x, by the action of the body M, very nearly as if all the bodies of
the system were collected into one at the center. The same result evi
dently takes place relatively to the axes of y and z; so that the forces, by
B2
20 A COMMENTARY ON [SECT. XL
which the center of gravity of the system is animated parallel to these
axes, by the action of M, are respectively
My Mz
"6);a '"&''
When we consider the relative motion of the center of gravity of the
system about M, the direction of the force which sollicits M must be
changed. This force resulting from the action of (*, p, &c. upon M, and
resolved parallel to x, in the contrary direction from the origin, is
if we neglect small quantities of the second order, this function becomes,
after what has been shown, equal to
X 2./C6
I3
In like manner, the forces by which M is actuated arising from the
system, parallel to the axes of y, and of z, in the contrary direction, are
It is thus perceptible, that the action of the system upon the body M,
is very nearly the same as if all the bodies were collected at their common
center of gravity. Transferring to this center, and with a contrary sign,
the three preceding forces; this point will be solicited parallel to the
axes of x, y and z, in its relative motion about M, by the three following
forces, scil.
- (M + ?(*) _(M + 2^) -~y _ (M + 2.
(sr (s)3 (s)3
These forces are the same as if all the bodies /«•, ft', /*", &c. were col
lected at their common center of gravity; which center, therefore, moves
nearly (to small quantities of the second order] as if all the bodies were col
lected at that center.
Hence it follows, that if there are many systems, whose centers of gra
vity are very distant from each other, relatively to the respective distances
of the bodies of each system ; these centers will be moved very nearly, as
if the bodies of each system were there collected ; for the action of the
first system upon each body of the second system, is the same very nearly
as if the bodies of the first system were collected at their common center
of gravity ; the action of the first system upon the center of gravity of the
second, will be therefore, by what has preceded, the same as on this hy
pothesis ; whence we may conclude generally that the reciprocal action of
BOOK L] NEWTON'S PRINCIPIA. 21
different systems upon their respective centers of gravity > is the same as if all
the bodies of each system were there collected, and also that these centers
move as on that supposition.
It is clear that this result subsists equally, whether the bodies of eacli
system be free, or connected together in any way whatever ; for their mu
tual action has no influence upon the motion of their common center
of gravity.
The system of a planet acts, therefore, upon the other bodies of the
Solar system, very nearly the same as if the Planet and its Satellites,
were collected at their common center of gravity; and this center itself is
attracted by the different bodies of the Solar system, as it would be on
that hypothesis.
Having given the equations of motion of a system of bodies submitted
to their mutual attraction, it remains to integrate them by successive
approximations. In the solar system, the celestial bodies move nearly as
if they obeyed only the principal force which actuates them, and the per
turbing forces are inconsiderable; we may, therefore, in a first approxi
mation consider only the mutual action of two bodies, scil. that of a planet
or of a comet and of the sun, in the theory of planets and comets ; and
the mutual action of a satellite and of its planet, in the theory of satellites.
We shall begin by giving a rigorous determination of the motion of two
attracting bodies : this first approximation will conduct us to a second in
which we shall include the first powers of small quantities or the perturb
ing forces ; next we shall consider the squares and products of these
forces; and continuing the process, we shall determine the motions of the
heavenly bodies with all the accuracy that observations will admit of.
FIRST APPROXIMATION.
478. We know already that a body attracted towards a fixed point,
by a force varying reciprocally as the square of the distance, de
scribes a conic section ; or in the relative motion of the body p, round
M, this latter body being considered as fixed, we must transfer in a di- '
rection contrary to that of p, the action of/* upon M; so that in this re
lative motion, p is solicited towards M, by a force equal to the sum ol "
the masses M, and i* divided by the square of their distance. All this
has been ascertained already. But the importance of the subject in the
Theory of the system of the world, will be a sufficient excuse for repre
senting it under another form,
B3
22 A COMMENTARY ON [SECT. XL
First transform the variables x, y, z into others more commodious for
astronomical purposes, g being the distance of the centers of p and M,
call (v) the angle which the projection of g upon the plane of x, y makes
with the axis of x; and (6) the inclination of g to the same plane; we
shall have
x = f cos. 6 cos. v ; -\
y = g cos. 6 sin. v; V ........ (1)
z = g sin. 6. }
Next putting
we have
/dQ
M + ,
*'o
ex' + yy'+ zz7)
e
!\ 1 /d Xx
M+/
M
f/3
dx
1/1^
2 . 7 3
^X
Similarly
Q\ _ 1 /dj^x M
~
^x _ _ v ftz
/ ~ g3 ' g3 '
Hence equations (1), (2), (3) of number 471, become
d2x /dQx d^y /d_Qx d'z _ /dQv
dt2 ' \dx-/; dt1 ~: \dy)' d t2~ VdzJ '
Now multiplying the first of these equations by cos. 6. cos. v; the
second by cos. 6. sin. v ; the third by sin. 6, we get, by adding them
In like manner, multiplying the first of the above equations by — g cos.0 X
sin. v; the second by g cos. 6 cos. v and adding them, &c. we have
inr
And lastly multiplying the first by — g sin. 6. cos. v ; the second by
— g sin. 6. cos. v and adding them to the third multiplied by cos. 6. we
have
To render the equations (2), (3), (4), still better adapted for use, let
1
u ~
g cos. 0
BOOK I.]
NEWTON'S PRINCIPIA.
23
and
s = tan. 6
u being unity divided by the projection of the radius g upon the plane
of x, y ; and s the tangent of the latitude of (A from that same plane.
If we multiply equation (3) by gz d v cos.2 6 and integrate, we get
h being the arbitrary constant.
Hence
d t =
d
j
d
(5)
^ '
\ u
If we add equation (2) multiplied by — cos. 6 to equation (4) multi
plied by — - — , we shall have
d«i
u
1 d
whence
r d u \ d v £ . , f /d Q\ s /d Q\ 1
.^n)+^r, = "Qdt{(^) + 7r(ds)}-
Substituting for d t, its foregoing value, and making d v constant, we
shall have
o77 nr — .... (6)
0 =
d v
d v / u 2
In the same way making d v constant, equation (4) will become
dQ
0 =
() _/)
d * s- . . d v \ d v / \ d u/ v ;
d s
d v
. . . (7)
Now making M + ^ = m, we have (in this case)
f-^ m m u
Q = — or = — r-r- r
g V (1 + s2)
and the equations (5), (6), (7) will become
dv
dt =
h.u
0=^ + u_
hs(l
0 =
(8)
24 A COMMENTARY ON [SECT. XI.
(These equations may be more simply deduced directly 124 and Wood-
house's Phys. Astron.)
The area described during the element of time d t, by the projection
of the radius-vector is £ i-? ; the first of equations (8) show that tins area
is proporti&nal to that element, and also that in a finite time it is propor
tional to the time.
Moreover integrating the last of them (by 122) or by multiplying by
2 d s, we get
s = y sin. (v — 0) ......... (9)
7 and 6 being two arbitrary constants.
Finally, the second equation gives by integration
U = h*(l+V) ^1 + S" + ecos.(v-w)} = V1+s"; . . . (10)
e and -a being two new arbitraries.
Substituting for s in this expression, its value in terms of v, and then
this expression in the equation
the integral of this equation will give t in terms of v ; thus we shall have
v, u and s in functions of the time.
This process may be considerably simplified, by observing that the
value of s indicates the orbit to lie wholly in one plane, the tangent of
whose inclination to a fixed plane is 7, the longitude of the node 6 bein^
reckoned from the origin of the angle v. In referring, therefore, to this
plane the motion of//,; we shall have
s = 0 and 7=0,
which give
] p
u = ? = pU + ecos- (v — «OJ.
This equation is that of an ellipse in which the origin of g is at the
focus :
is the semi-axis-major which we shall designate by a; e is the ratio of
the excentricity to the semi-axis-major ; and lastly * is the longitude of
the perihelion. The equation
d v
d t = _
h u
BOOK L] NEWTON'S PRINCIPIA. 25
hence becomes
d - a^(l — e2)^ ' d v
V^fJi, {1+ecos. (v — •&)}"'
Develope the second member of this equation, in a series of the angle
v — -a and of its multiples. For that purpose, we will commence by
developing
1
1 + e cos. (v — w)
in a similar series. If we make
X =
1 + V ( 1 — e 2) '
we shall have
1 1_ _f 1 X. c-(v-*Q^-
1+ecos. (v — w)~ y'l e* I l + Xc(v~w) — 1 1 + Xc — (v— ^OV
e being the number whose hyperbolic is unity. Developing the second
member of this equation, in a series; namely the first term relatively
to powers of c(v~ w)v/— i1, and the second term relatively to powers of
c ~ (v — **") y— l and then substituting, instead of imaginary exponentials,
their expressions in terms of sine and cosine ; we shall find
I + e cos. (v — -af] " V 1 e2
{I — 2 X cos. (v — w) + 2 X2 cos. 2 (v — »)— 2 X 3 cos. 3 (v — *) + &c.|
Calling <p the second member of this equation, and making q = — ; we
shall have generally
i = ±e" "ldm'(T
? 1 + e coa, (v — ~)}m + L 1.2.3 in. d q M '
for putting
q ' q + R
R being = cos. (v — w)
*-e)
1
(q + R)2
**'(j) _^
dq2 -(q + il)«
&c. = &c.
26 A COMMENTARY ON [SECT. XI.
___ 4- 2- 3 ____ m
~~~
— (q
_
dqra 2.3...m~ (q+ R)m + '
1
£1 + e cos. (v — w)Jm + 1"
Hence it is easy to conclude that if we make
U + e cos. (v — w)f = (* — e 2)
[I + E »). cos. (v — ,) + E (2lcos. 2 (v — ~) + &c.|
we shall have generally whatever be the number (i)
(1 + V 1 — e2)1
the signs + being used according as i is even or odd ; supposing there
fore that u r= a~ a V m, we have
ndt = dv [I + E(1>cos. (v — ») + E(2> cos. 2 (v — »)+ &c.*
and integrating
n t +e = v + E (1) sin. (v — ») + \ E (2) sin. 2 (v — ») + &c.
e being an arbitrary constant. This expression for n t + £ is very con
vergent when the orbits are of small excentricity, such as are those of the
Planets and of the Satellites ; and by the Reversion of Series we can find
v in terms of t : we shall proceed to this presently.
474. When the Planet comes again to the same point of its orbit, v is
augmented by the circumference 2 it ; naming therefore T the time of the
whole revolution, we have (see also 159)
T-i-'-^.
n V m
This could be obtained immediately from the expression
T1 — J °
~TT~
2 area of Ellipse _ 2jra b
~~h~ IT'
But by 157
h s = m a ( 1 — e2)
2_ „ 2
_ it a
V m
BOOK I.]
NEWTON'S PRINCIPIA.
27
If we neglect the masses of the planets relatively to that of the sun we
have
which will be the same for all the planets j T is therefore proportional in
that hypothesis to a 2, and consequently the squares of the Periods are as
the cubes of the major axes of the orbits. We see also that the
same law holds with regard to the motion of the satellites around their
planet, provided their masses are also deemed inconsiderable compared
with that of the planet.
475. The equations of motion of the two bodies M and fj<> may also be
integrated in this manner.
Resuming the equations (1), (2), (3), of 471, and putting M + /* = m, we
have for these two bodies
0 =
0 ~
0 =
dt2
!!_?
dt2
d2z
dt2
x m x
• - 3
m y
m z
(0)
The integrals of these equations will give in functions of the time t, the
three coordinates x, y, z of the body & referred to the center of M ; we
shall then have (471) the coordinates £, n, 7 of the body M, referred to a
fixed point by means of the equations
- a
bt —
* x
m
H = a7 + b7 t —
= a" + b"t —
m
Lastly, we shall have the coordinates of ^ referred to the same fixed
point, by adding x to £, y to n, and z to y : We shall also have the rela
tive motion of the bodies M and /«*, and their absolute motion in space.
476. To integrate the equations (0) we shall observe that if amongst
the (n) variables x^, x ^2) x (n) and the variable t, whose difference
is supposed constant, a number n of equations of the following form
0 = — f
H . X
dt' dt1- dt»-
in which we suppose s successively equal to 1, 2, 3 n ; A, B H
oeing functions of the variables x (1), x (2), &c. and of t symmetrical
28
A COMMENTARY ON
[SECT. XL
with regard to the variables x (1), x (2>, &c. that is to say, such that they
remain the same, when we change any one of these variables to any other
and reciprocally ; suppose
x (1) __ a (I) x (n - i + 1) _j_ b (1) x (x - i + 2) _f_ h (1) x (n) ?
X (2) = a (2) x(n-i + l) _|_ b (2) x (n-i + 2) _^. h (2) X n.
x — a 'X ~T~" x ~T" ft ' x.''
a (1), b (n, h (1) ; a (2), b (2), &c. being the arbitraries of which the
number is i (n — i). It is clear that these Values satisfy the proposed
system of equations : Moreover these equations are thereby reduced to i
equations involving the i variables x(n~i + 1) xw. Their integrals
will introduce i 2 new arbitraries, which together with the i (n — i) pre •
ceding ones will form i n arbitraries which ought to give the integration
of the equations proposed.
477. To apply the above Theorem to equations (0) ; we have
z = a x + b y
a and b being two arbitrary constants, this equation being that of a plane
passing through the origin of coordinates ; also the orbit of ^ is wholly in
one plane.
The equations (0) give
(0')
g • — x 2 + y 2 + z
Also since
and
and differentiating twice more, we have
and consequently
+ 3(dxd2x + dyd2y + dzd2z),
d 3 x d 3 y d 3
d2x . d2v .
Substituting in the second member of this equation for d 3 x, d 3 y, d • z
BOOK L] NEWTON'S PRJNCIPIA. 29
their values given by equations ((X), and for d 2 x, d 2 y, d 2 z their values
given by equations (0) ; we shall find
If we compare this equation with equations (0'), we shall have in virtue
of the preceding Theorem, by considering -y — , -^ , -.- , y|, as so many
particular variables x (1), x (2\ x (3), x W, and g as a function of the time t;
d g = A d x + y d y ;
X and 7 being constants ; and integrating
h2
= -. + Xx + 7y,
h 2
— being a constant. This equation combined with
z = ax + by;g2 = x2 + y2 + z*
gives an equation of the second degree in terms of x, y, or in terms of
x, z, or of y, z; whence it follows that the three projections of the curve
described by p about M, are lines of the second order, and therefore that
the curve itself (lying in one plane) is a line of the second order or a conic
section. It is easy to perceive from the nature of conic sections that, the
radius-vector g being expressed by a linear function of x, y, the origin of
x, y ought to be in the focus. But the equation
h2
e = m + Xx + yy
gives by means of equations (0)
( V\
_ d2g , V mJ
° = d^ + ^- p-
Multiplying this by d g and integrating we get
a' being an arbitrary constant. Hence
dt=
I /« e » \
m J (2 g — -r )
V v a' m /
which will give g in terms of t ; and since x, y, z are given above in terms
of g, we shall have the coordinates of ^ in functions of the times.
478. We can obtain these results by the following method, which has
the advantage of giving the arbitrary constants in terms of the coordinates
x, y, z and of their first differences ; which will presently be of great use
to us.
30 A COMMENTARY ON [SECT. XI.'
Let V = constant, be an integral of the first order of equations (0), V
being a function of X, y, z, -, - , -r^ , i-? . Call the three last quantities
x', y', z'. Then V = constant will give, by taking the differential,
/d Vx dx ,d V, d y ,dVx d z
' VdxV ' dt " VdyJ ' dt " Vdz ' ' dt
r(LY\ dx/ j fdJ^ d y' fd v d z/
" Vd xV ' dt " " Vd yV ' dt" " " \d z') ' "dt"
But equations (0) give
d x' m x d y' m y d z' m z
dT : ' ~7r ' "dT : ' ~7»' ' dT : "~p~ ;
we have therefore the equation of Partial Differences
,
0 = x
, /d Vx
(a)
m /dV
It is evident that every function of x, y, z, xr, y'. z' which, when sub
stituted for V in this equation, satisfies it, becomes, by putting it equal to
an arbitrary constant, an integral of the first order of the equations (0).
Suppose
V = U + U' + U" + &c.
U being a function of x, y, z; U' a function of x, y, z, x', y', z' but of the
first order relatively to x', y', z' ; U" a function of x, y, z, xx, y', z' and of
the second order relatively to x', y', z', and so on. Substitute this value
of V in the equation (I) and compare separately 1. the terms without
x', y', z' ; 2. those which contain their first powers ; 3. those involving their
squares and products, and so on ; and we shall have
U'x /d U'x /d U'
m f /d U"x /d U
= x
, /d U'x , ,/d U\ , 7/d U'x m / /d Uwx w/dU
x "-
m
&c.
which four equations call (F).
The integral of the first of them is
U' = ftmct. Jx y' — y x', x z' — z x', y z' — z y', x, y, z]
BOOK I.] NEWTON'S PRINCIPIA. 31
The value of U7 is linear with regard to x7, y7, z7 ; suppose it of this
form
U' = A (x y' — y x7) + B (x z7 — z x') + C (y z' — z y7) ;
A, B, C being arbitrary constants. Make
U"7, &c. = 0;
then the third of the equations (F) will become
The preceding value of (J7 satisfies also this equation.
Again, the fourth of the equations (F) becomes
of which the integral is
U7' = funct. £x y7 — y x7, x z7 — z x7, y z7 — z y7, x7, y7, z7}.
This function ought to satisfy the second of equations (F), and the first
member of this equation multiplied by d t is evidently equal to d U. The
second member ought therefore to be an exact differential of a function of
x, y, z ; and it is easy to perceive that we shall satisfy at once this condi
tion, the nature of the function U77, and the supposition that this function
ought to be of the second order, by making
U77 = (D y7 — E x') . (x y7 — y x7) + (D z7 — F x7) (x z7 — z x7)
+ (E z' — F y') (y z' — z y') + G (x'2 + y 2 + z' 2)j
D, E, F, G being arbitrary constants ; and then g being = V7x2+y2+z2,
we have
U = --m(Dx + Ey+ Fz + 2G);
Thus we have the values of
U, U', U" ;
and the equation V = constant will become
const. =— m£D x+E y+F z + 2 G} + (A + D y' — E x7) (x y' — y X')
+ (B + D z' — F x') (x z' — z x7) + (C + E z7 — F y) (y z' — z y)
+ G (x 2 + y 2 + z 2).
This equation satisfies equation (I) and consequently the equations (0)
whatever may be the arbitrary constants A, B, C, D, E, F, G. Sup
posing all these = 0, 1. except A, 2. except B, 3. except C, &c. and
putting
d x d y d z „
d!' dt' t,lfor*',y,z,
A COMMENTARY ON [SECT. XI.
we shall have the integrals
(P)
c — —
d y > d x fl x
d
z —
zdx
, C/X
dy
j d z
— zd
y
dt
, V, —
d y 2 ~K
d
dt
z2)
1 y
.d
x
d t
z d z .
dx
0 = f
0 f"
\ s
-1 yfm
dx
2 +
d
z2)
1 x
d t
dx
.d
y .
z d z.
dy
1 } If
, r m
dx
dt2
2 +
d
J
y^
T
1 x
dt
d x
2
.d
i
z |
dt
y dy.
i
dz
n-™
j- x
•d+'
1
T^
2dl
T^
dt
c, c', c", f, f', f" and a being arbitrary constants.
The equations (0) can have but six distinct integrals of the first order,
by means of which, if we eliminate d x, d y, d z, we shall have the three
variables x, y, z in functions of the time t; we must therefore have at least
one of the seven integrals (P) contained in the six others. We also per
ceive d priori, that two of these integrals ought to enter into the five
others. In fact, since it is the element only of the time which enters
these integrals, they cannot give the variables x, y, z in functions of the
time, and therefore are insufficient to determine completely the motion of
«• about M. Let us examine how it is that these integrals make but five
distinct integrals.
Z Q V '^ __ V d Z
If we multiply the fourth of the equations (P) by -- *—^ — *• - , and
Y H z ___ Z Cl X
add the product to the fifth multiplied by - — j— — - , we shall have
n— f zdy — ydz f, xdz — zdx xdy — y dx f m d x2-f d y2)
~~dT~ ~dT~ "Tt~ IT" dt2 j
xdy — ydxfxdx. dz
~d~r ~ i ~~d T 2~
d.dz
yy
d
„ xdy — ydx xdz — zdx ydz — zdy , .
Substituting for - dt - » - g~j: - > - 3~£ - - » their
values given by the three first of the equations (P), we shall have
f ' c' — f c" ( m d x2 + d yg \ x d x . d z y d y.d z
~^~ Z\7" "dT^" J dte ~~d"t^~
This equation enters into the sixth of the integrals P, by making
f" = f/ c/ ~ f c" or 0 = f c" — f c' + f" c. Also the sixth of these
c
integrals results from the five first, and the six arbitraries c, c', c'7, f, f, f"
are connected by the preceding equation.
BOOK I.] NEWTON'S PRINCIPIA. 33
If we take the squares off, P, f" given by the equations (P), then add
them together, and make f 2 + P 2 -f- P' 2 = 1 2, we shall have
-/0* dx2+dy2+dz2
+ dy2+dz' gmi
d~t2 ~J''
dts Vdt/ J * \ d~
but if we square the values of c, c', c", given by the same equations, and
make c2 + c/2 + c"2 = h2; we get
dt2
the equation above thus becomes
d x2+ dy2+ d z2 2m m2 — I2
dt2 "7" " h2 *
Comparing this equation with the last of equations (P), we shall have
the equation of condition,
m2 — I2 _m
h* a *
The last of equations (P) consequently enters the six first, which are
themselves equivalent only to five distinct integrals, the seven arbitrary
constants, c, c', c", f, P, f", and a being connected by the two preceding
equations of condition. Whence it results that we shall have the most
general expression of V, which will satisfy equation (I) by taking for this
expression an arbitrary function of the values of c, c', c", f, and P, given
by the five first of the equations (P).
479. Although these integrals are insufficient for the determination of
x, y, z in functions of the time ; yet they determine the nature of the
curve described by ft about M. In fact, if we multiply the first of the
equations (P) by z, the second by — y, and the third by x, and add the
results, we shall have
0 = c z — c' y -f- c" x,
the equation to a plane whose position depends upon the constants
c c' c"
c, c , c .
If we multiply the fourth of the equations (P) by x, the fifth by y, and
the sixth by z, we shall have
but by the preceding number
, dx2+ dy*+ dz2
dt2 dt2
.-. 0 = m g — h2 + f x + f7 y + f" z.
This equation combined with
0 = c" x — c' y + c z
VOL. II. C
34 A COMMENTARY ON [SECT. XI.
and
g« = x2 + y2 + z2
gives the equation to conic sections, the origin of § being at the focus.
The planets and comets describe therefore round the sun very nearly
conic sections, the sun being in one of the foci ; and these stars so move
that their radius-vectors describe areas proportional to the times. In fact,
if d v denote the elemental angle included by £, g + d f, we have
d x8 + d y 2 + d z 2 = s z d v 2 + d s 2
and the equation
dt2 dt2
becomes
*4d v2 = h2d t1;
hdt
,.dv=_.
Hence we see that the elemental area I § 2 d v, described by f, is propor
tional to the element of time d t ; and the area described in a finite time is
therefore also proportional to that time. We see also that the angular
motion of ^ about M, is at every point of the orbit, as — -z ; and since 'without
sensible error "we may take very short times for those indefinitely small, we
shall have, by means of the above equation, the horary motions of the planets
and comets, in the different points of their orbits.
The elements of the section described by p, are the arbitrary constants
of its motion ; these are functions of the arbitraries c, c', c", f, P, f", and
— . Let us determine these functions.
a
Let 6 be the angle which the intersection of the planes of the orbit and
of (x, y) makes with the axis of x, this intersection being called the line
of the nodes ; also let <p be the inclination of the planes. If x', y' be the
coordinates of //. referred to the line of the nodes as the axis of abscissas,
then we have
x' = x cos. 6 + y sin. 6
• y' = y cos. d — x sin. 6.
Moreover
z = y' tan. <p
.: z = y cos. 6 tan. <p — x sin. 6 tan. <p.
Comparing this equation with the following one
0 = c" x — c' y + c /
BOOK L] NEWTON'S PRINCIPIA. 35
we shall have
c' = c cos. 6. tan. p
c" = c sin. 6 tan. <p
whence
c"
tan. d = —
c
and
t,n.?= V(c"jL0
C
Thus are determined the position of the nodes and the inclination of the
orbit, in functions of the arbitrary constants c, c', c".
At the perihelion, we have
g d g = 0, orxdx + ydy + zdz — 0.
Let X, Y, Z be the coordinates of the planet at this point ; the fourth
and the fifth of the equations (P) will give
_Y _ P
-A. I
But if I be called the longitude of the projection of the perihelion upon
the plane of x, y this longitude being reckoned from the axis of x, we have
Y
v- = tan. I ;
which determines the position of the major axis of the conic section.
If from the equation
dx«+ dy2 + dZ* g*dg
d t2 d t2
,. . dx2 + d y2 + d z2
we eliminate - - — p^ - , by means of the last of the equa
tions (P), we shall have
but d § is 0 at the extremities of the axis major ; we therefore have at these
points
„ v, 2
0 - e*_2a S+ ---
m
The sum of the two values of g in this equation, is the axis major, and
their difference is double the excentricity ; thus a is the semi-axis major of
the orbit, or the mean distance of p from M ; and
36 A COMMENTARY ON [SECT. XI.
is the ratio of the excentricity to the semi-axis major. Let
/y ^ ma/
and having by the above
m _ m 2 — 1 2
"a" "IT2 ;
we shall get
m e = 1.
Thus we know all the elements which determine the nature of the conic
section and its position in space.
480. The three finite equations found above between x, y, z and g give
x, y, z in functions of g ; and to get these coordinates in functions of the
time it is sufficient to obtain g in a similar function ; which will require a
new integration. For that purpose take the equation
f ,
-- =•_ — h
dt2
But we have above
h2 = -- (m2 — I2) = am (1 — e2);
P d P
... d t = L-J _ .
V m I 12 g — S- a (1 — e2) V
whose integral (237) is
t + T = ~ — (u — e sin. u) (S)
/I s \
u being = cos.—1 f )' an<^ ^ an arbitrary constant.
This equation gives u and therefore g in terms of t; and since x, y, z
are given in functions of g, we shall have the values of the coordinates for
any instants whatever.
We have therefore completely integrated the equations (0) of 475, and
thereby introduced the six arbitrary constants a, e, I, 6, <p, and T. The
two first depend upon the nature of the orbit ; the three next depend upon
its position in space, and the last relates to the position of the body u.
at any given epoch ; or which amounts to the same, depends upon the
instant of its passing the perihelion.
Referring the coordinates of the body ^, to such as are more commodious
for astronomical uses, and for that, naming v the angle which the radius-
BOOK I.] NEWTON'S PRINCIPIA. 37
vector makes with the major axis setting out from the perihelion, the
equation to the ellipse is
a (1 — e2)
1 + e cos, v '
The equation
g = a ( 1 — e cos. u)
indicates that u is 0 at the perihelion, so that this point is the origin of two
angles u and v ; and it is easy hence to conclude that the angle u is formed by
the axis major, and by the radius drawn from its center to the point where
the circumference described upon the axis major as a diameter, is met by
the ordinate passing through the body p at right angles to the axis major.
Hence as in (237) we have
v 1 1 + e _ u
tan- 2 = ^T=-e'taU'-2'
We therefore have (making T = 0, &c.)
n t = u — e sin. u
§ = a ( 1 — e cos. u)
and
v / 1 + e u
(0
n t being the Mean Anomaly,
n the Excentric Anomaly,
v the True Anomaly.
The first of these equations gives u in terms of t, and the two others
will give g and v when u shall be determined. The equation between u
and t is transcendental, and can only be resolved by approximation.
Happily the circumstances attending the motions of the heavenly bodies
present us with rapid approximations. In fact the orbits of the stars are
either nearly circular or nearly parabolical, and in both cases, we can de
termine u in terms of t by series very convergent, which we now proceed
to develope. For this purpose we shall give some general Theorems
upon the reduction of functions into series, which will be found very use
ful hereafter.
481. Let u be any function whatever of «, which we propose to deve
lope into a series proceeding by the powers of a. Representing this
series by
U = «> a.q,+ as.cj8+ a". qn + an + | . qD+ ' + &c.
C3
38 A COMMENTARY ON [SECT. XL
"j qi> q-25 &c- being quantities independent of a, it is evident that u is what
u will become when we suppose a = 0 ; and that whatever n may be
= 1.2....n.qn + 2.3....(n+l).a.qn + 1 + &c.
/d n u\
the difference ( ~ — -J being taken on the supposition that every thing in
u varies with a. Hence if we suppose after the differentiations, that a = 0,
, . /dnu\
in the expression (-, — -J we have
dnu\
X
1.2 ____ n'
This is Maclaurin's Theorem (see 32) for one variable.
Again, if u be a function of two quantities a, a', let it be put
u = U + a . q1)0 + a 2 . q2;0 + &C.
+ <*'. qo,i + ««'• qi,i + &c.
+ «'2- qo,2 + &c.
the general term being
«tta'n'qn.n'
Then if generally
/ d n + n' u
\d «n. d u'n'
denotes the (n + n')th difference of u, the operation being performed (n)
times, on the supposition that a is the only variable, and then n' times on
that of a! being the only variable, we have
a 2' q3>0 "*" 4 a 3 q4>0 + 5 a*' q5>0 + &c'
.a/q2,i +3a2a'q3jl +4a3a/q4>1 + &C.
a/222 +3asa/« + &c.
' 2 a ^.° + 4< 3 a 2 ^.o + 5. 4 a3 q5>0 + &c.
2 « q2>, + 3. 2aaq3>1 + 4. 3a2aq4jl + 8cc.
+ 2 a2 2 3.2aa2 &C.
T&) = 2 q2)1 + 3. 2 a q3)1 + &c.
+ 2 a q2>2 + &c.
and continuing the process it will be found that
•Tjr = 2- 3- . . . n X 2. 3. . ..n' X q,n,
BOOK I.] NEWTON'S PRINCIPIA. 39
so that when «, a both equal 0, we have
/ d B + "' u N
Vd «°.d «'".'/
q"'n/ -27^77.. n x 2.3....n' ^
And generally, if u be a function of a, d, a", &c. and in developing it
into a series, if the coefficient of « ». & "'. «" n". &c. be denoted by qn, n,, „., &c
we shall have, in making «, d, a", &c. all equal 0,
(c] n + n' + n" + &c. u
_ d«".dd"'.da""",&C.)
] ;" "> ""' - 2.3....nX-273- . . . n' X 2. 3 . . . . n" X &c. ' ' ' (2)
This is Maclaurin's Theorem made general.
482. Again let u be any function of t + «, t' + «, t" + a", &c. and
put
u = g> (t + a, t' + a, t" + a", &c.)
then since t and a are similarly involved it is evident that
d n + "' + "" + &c- . u \ __ / dn + n' + n" + &c- . u \
Vd an. d a. n/. d «//n"~&c"./ = \d tn. d t/n'. d t" n". &c./
and making
«, A, a", &c. = 0,
or
u = <p (t, t7, t", &c.)
by (2) of the preceding article we have
, t', t", &c.)x
d t n. d t/n<. d t" n" &c /
. .
ln'n/>n"'&c ~ 2.3.. ..n X 2. 3 . . . . n
which gives Taylor's Theorem in all its generality (see 32).
Hence when
u = <f> . (t + «)
dn.?(t)
" 2.3 ____ n.dt"
and we thence get
»(« + .) = > (t) + .!^fil + "_!.^ + &c ...... (i)
483. Generally, suppose that u is a function of «, «, a'x, &c. and of
tj tr, t", &c. Then, if by the nature of the function or by an equation of
R. ftial Differences which represents it, we can obtain
/ d n + n/ + &c- . u v
Vda". da"'. &cJ
in a function of u, and of its Differences taken with regard to t, t', &c.
40 A COMMENTARY ON [SECT. XI.
calling it F when for u we put u or make a, a, a", &c. = 0 ; it is evident
we have
_ F _
qn.n'.n-.to. ~ g. 3 . . . n X g. 3 . . . n' X 2. 3 . . . n", X &C.
and therefore the law of the series into which u is developed.
For instance, let u, instead of being given immediately in terms of a,
and t, be a function of x, x itself being deducible from the equation of
Partial Differences
in which X is any function whatever of x. That is
Given
u = function (x)
d
to develop e u into a series ascending by the powers of a.
First, since
/dux P/d/Xd_Ux
• • Vd a) ~ \ d t )
Hence
_
a*)~ \ da.dt J'
But by equation (k), changing u into J X d u
,d./Xdux _ /d./X2dux
v do )~\ dt ;•
. fd'u\ _ /d2/X2dux
' V d aV ~ \ d t 8 /
Again
/d3ux __ /d3/X2dux
\da3J-l da.dt2 /'
But by equation k, and changing u into f X 2 d u
/d/X*dux _ /d/X3dux
\ ' d« y~v dt J
/d'uv /d3./^^_d_ux
••Vd^v-V dt3 r
Thus proceeding we easily conclude generally that
Now, wlien a = 0, let
x = function of t = T
m
BOOK I.] NEWTON'S PRINCIPIA. 41
and substitute this value of x in X and u ; and let these then become X
and u respectively. Then we shall have
.
/cPMiN =
\dan/ d t"-1
and
" • A ' d T~ /0.
•'* q" -"2737 .ndt»-1 (2)
which gives
, .
du , a2 >> d t/ , a
-d-t + T. — dl — + — ..
which is Lagrange's Theorem.
To determine the value of x in terms of t and a, we must integrate
In order to accomplish this object, we have
and substituting
we shall have
d x = \d t + X d .;
( x\
i ^d t /
.-. d x = ^
which by integration, gives
x = p (t + a X) . (2)
<p denoting an arbitrary function.
Hence whenever we have an equation reducible to this form x =
f (t + « X), the value of u will be given by the formula (p), in a series of
the powers of a.
By an extension of the process, the Theorem may be generalized to the
case, when
u = function (x, x', x", &c.)
42 A COMMENTARY ON [SECT. XI.
and
x = p (t + a X)
x' = ?' (t' + «' X')
x" = p" (t" + a" X")
&c. = &c.
484. Given (237)
u — n t + e sin. u
required to develope u or any Junction of it according to ike powers ofe.
Comparing the above form with
X = ? (t + a X)
x, t, a, X become respectively
u, n t, e, sin. u.
Hence the formula (p) 483. gives
e2 d H/(nt)sin.8nt*
+ (u) = -4,(nt) + e -V (n t) sin. n t + - . - ~^—
e3 d2£-4/ (nt) sin.3nt} .
+ 2T3-- n2dt2 - + &C ........ W
V (n t) being = .
To farther develope this formula we have generally (see Woodhouse's
Trig.)
-
-
sin.i(nt) = - —^—^ - ; cos.'(nt) =
c being the hyperbolic base, and i any number whatever. Developing the
second members of these equations, and then substituting
cos. r n t + V — I sin. r n t, and cos. r n t — V — 1 sin. r n t
for crnt ^""S and c~rn t\/~1., r being any number whatever, we shall
have the powers i of sin. n t, and of cos. n t expressed in shies and cosines
of n t and its multiples ; hence we find
e e2
P = sin. n t + -jj sin 2 n t + 5-5 sin. 3 n t + &c.
^ £* O
= sin. n t — 5^5 . {cos. 2 n t — 1 }
in' 5 n '-5 sin- 3 n t+TT2 sill> n
BOOK I.] NEWTON'S PRINCIPIA. 43
6.5 1 6.5.4
O Q A t; « 05' 1 w»?w«* "^o.^m-p— — ^.v — .
/i.O.<i.O.U.<w (_ l.-c <& l.iC.o
— &c.
Now multiply this function by -\J/ (n t), and differentiate each of its
terms relatively to t a number of times indicated by the power of e which
multiplies it, d t being supposed constant; and divide these differentials
by the corresponding power of n d t. Then if P7 be the sum of the
quotients, the formula (q) will become
4 (u) = -^ (n t) + e P'.
By this method it is easy to obtain the values of the angle u, and of
the sine and cosine of its multiples. Supposing for example, that
•^ u = sin. i u
we have
•4/ (n t) = i cos. int.
Multiply therefore the preceding value of P, by i. cos. i n t, and deve-
lope the product into sines and cosines of n t and its multiples. The
terms multiplied by the even powers of e, are sines, and those multiplied
by the odd powers of e, are cosines. We change therefore any term of
the form, K e°T sin. s n t, into + K e 2 r s 2 r sin. s n t, + or — obtaining
according as r is even or odd. In like mariner, we change any term
of the form, K e2r + l cos. s n t, into + K e2r+l. s2r + l. sin. s n t, — or
-f- obtaining according as r is even or odd. The sum of all these terms
will be P' and we shall have
sin. i u = sin. i n t + e P'.
But if we suppose
•4/ (u) = u;
then
>}/ (n t) = 1
and we find by the same process
e2
u = n t + e sin. n t -f- ~— ^ . 2 sin. 2 n t
e3
+ .{3 2 sin. 3 n t — 3 sin. n t}
a 4
. [4, 3 sin. 4 n t — 4.2s sin. 2 n t}
e s f 54
34524- |54sin.5nt — 5. 34sin. 3 n t+^sin. n 1
&c.
44 A COMMENTARY ON [SECT. XL
a formula which expresses the Excentric Anomaly in terms of the Mean
Anomaly.
This series is very convergent for the Planets, Having thus determin
ed u for any instant, we could thence obtain by means of (237), the cor
responding values of f and v. But these may be found directly as fol
lows, also in convergent series.
485. Required to express g in terms of the Mean Anomaly.
By (237) we have
§ = a (1 — e cos. u).
Therefore if in formula (q) we put
•^ (u) = 1 — e cos. u
we have
y (n t) = e sin- n t,
and consequently
e 3 d sin. 3 n t
1 — e cos. u = 1 — e cos. n t + e 2 sin. 2 n t + — . • j- H &c.
- 11(11
Hence, by the above process, we shall find
P e * e *
-=-=!+ — — e cos. n t — cos. 2 n t
a <* »
e3
— • . [3 cos. 3 n t — 3 cos. n t }
<&• fit
.£42 cos. 4 n t — 4. 22. cos. 2 n t}
2. 3. 2 3
_ e* . 1 5 3 cos. 5 n t— 5. 3 3 cos. 3 n t + ^|. cos. u t j
_ e& 5 { 64cos. 6 nt— 6. 4 4 cos. 4 n t+^|. 2* cos.2nt [
*w»O»'i1«O«/i>. 1 • iw J
— &c.
486. To express the True Anomaly in terms of the Mean.
First we have (237)
Sin'lT ,1+e Sin'i
^r —V i — e* u
cos. -g- cos. -g
.•. substituting the imaginary expressions
CW
and making
-i— l /1 + e c"^-1— 1.
-1 + 1 — V i— e cuv-i+ 1*
__ e _
X~ 1 + V (1— e')
BOOK I.] NEWTON'S PRINCIPIA. 45
we shall have
1 __ v r — . u V — 1
cvV — i — cuV—
and therefore
whence expanding the logarithms into series (see p. 28), and putting-
sines and cosines for their imaginary values, we have
2 X 2 2 X 3
v = u + 2 X sin. u -| — ^— sin. 2 u -j -- ^— sin. 3 u + &c.
f£ o
But by the foregoing process we have u, sin. u, sin. 2 u, &c. in series
ordered by the powers of e, and developed into sines and cosines of n t
and its multiples. There is nothing else then to be done, in order to
express v in a similar series, but to expand X into a like series.
The equation, (putting u = 1 + V 1 — e2)
u
will give by the formula (p) of No. (483)
1 l i-e2 ,i(i + 8) e* i (i + 8) (i + 5) e6
~i -glT* "2T+T+ —3— '•2l+4"i 2.3 "<i 6a'
and since
u = 1 + V I —
we have
These operations being performed we shall find
e---j-e3 + j|j es| sin. n t
(103 451
+ I-96-6 -
1097
+ -960 6 S
1223 .
the approximation being carried on to quantities of the order e6 in
clusively.
46 A COMMENTARY ON [SECT. XL
487. The angles v and n t are here reckoned from the Perihelion ; but
if we wish to compute from the Aphelion, we have only to make e nega
tive. It would, therefore, be sufficient to augment the angle n t by «r, in
order to render negative the sines and cosines of the odd multiples of n t ;
then to make the results of these two methods identical ; we have only in
the expressions for g and v, to multiply the sines and cosines of odd
multiples of n t by odd powers of e ; and the even multiples by the even
powers. This is confirmed, in fact, by the process, a posteriori.
488. Suppose that instead of reckoning v from the perihelion, we fix
its origin at any point whatever ; then it is evident that this angle will be
augmented by a constant, which we shall call =>-, and which- will express
the Longitude of the Perihelion. If instead of fixing the origin of t at
the instant of the passage over the perihelion, we make it begin at any
point, the angle n t will be augmented by a constant which we will call
e — •»; and then the foregoing expressions for —and v, will become
a
•£ = 1 + 4-e2— (e— | e3)cos.(nt-H— »)— ( \ &—\ e4)cos.2(nt-H—
B 8 o £ o
where v is the true longitude of the planet and n t + l its mean longi
tude, these being measured on the plane of the orbit.
Let, however, the motion of the planet be referred to a fixed plane a
little inclined to that of the orbit, and <p be the mutual inclination of the
two planes, and 8 the longitude of the Ascending Node of the orbit, mea
sured upon the fixed plane ; also let $ be this longitude measured upon
the plane of the orbit, so that 6 is the projection of ft and lastly let v, be
the projection of v upon the fixed plane. Then we shall have
v, — 6, v — ft
making the two sides of a right angled spherical triangle, v — /3 being
opposite the right angle, and <p the angle included between them, and
therefore by Napier's Rules
tan. (v, — 6) = cos. <p tan. (v — /3) ...... (1)
This equation gives v, in terms of v and reciprocally ; but we can ex
press either of them in terms of the other by a series very convergent
after this manner.
By what has preceded, we have the series
11 X2 X3
- v = — u + X sin. u + ~ sin. 2 u + — sin. 3 u + &c.
BOOK I.] NEWTON'S PRINCIPIA 47
from
tan'2 v
by making
If we change - v into V/ — 69 and I u into v — ft and i~ t-? into
4 5» J — e
cos. p, we have
_ cos. p — 1 a
•« — -- — - — ~~ — rfin " _ «
cos. p + 1 If
The equation between - v and i u will change into the equation be
tween v, — 6 and v — ft and the above series will give
v,— 6 = V — /3- tan2- <p. sin. 2 (v — 8) + tan. 4 p. sin. 4 (v — /3)
— 3 tan. 6 - p sin. 6 (v — /3) + &c
If in the equation between | and ^ , we change ~ v into v _ /3 and
* /&
u into vy — tf, and , + 6 into — ~ , we shall have
^ —
e cos. <p
X = tan.2 -
and
v — /3 = v/_^ + tan. z ^ p. sin. 2 (v, — 0)
+ -jg tan. 4 - p. sin. 4 (vy — tf)
+ g tan. 6 - f . sin. 6 (v/ — tf) ..... (4)
Thus we see that the two preceding series reciprocally interchange,
ly changing the sign of tan. 2 £ p, and by changing v, — 6, v — j3 the (Tne
for the other. We shall have v/ - t in terms of the sine and cosine of
n t and its multiples, by observing that we have, by what precedes
v = n t + £ + e Q,
Q being a function of the sine of the angle n t + , - ., and its multi
ples; and that the formula (i) of number (482) gives, whatever is i,
sin. i (v — /3) = sin. i (n t + s — £ + e Q)
48 A COMMENTARY ON [SECT. XI.
Lastly, s being the tangent of the latitude of the planet above the fixed
plane, we have
s = tan. <p sin. (v, — 6} ;
and if we call fy the radius-vector projected upon the fixed plane, we
shall have
we shall therefore be able to determine v,, s and ^ in converging series
of the sines and cosines of the angle n t and of its multiples.
489. Let us now consider very excentric orbits or such as are those of
the Comets.
For this purpose resume the equations of No. (237), scil.
=
e cos. v
n t = u — e sin. u
tan. v =
In this case e differs very little from unity; we shall therefore suppose
1 — e = «
a being very small compared with unity.
Calling D the perihelion distance of the Comet, we shall have
D=u(l — e) = a a;
and the expression for g will become
_ - _ - ____>
" 2cos.*JI v — acos.v cos.2 i-
which gives, by reduction into a series
s =
cos.2 2
To get the relation of v to the time t, we shall observe that the expres
sion of the arc in terms of the tangent gives
u = 2 tan. i u {l - ± tan.2 \ u + \ tan.* I u - &c.}
But
1
BOOK I.] NEWTON'S PRINCIPIA. 49
\ve therefore have
« If 1 / a N 1 1 / « \ 2 * 1 1
u = 2 /'-- — tan.-v-J i — - (- Han.-— v-f ~ (^ ) tan.'^v — &c.f
-V 2 — a 21 3 \2 — a/ 2 5 \2 — a/ 2 )
Next we have
2 tan. — • u
sin. u =
1 + tan. 2 — u
A
i r
— = 2 tan. 4 u 1 1 — tan. 2 -^ + tan. 4 \ — &c.
1 25 f. A •&
Whence we get
/ «
-
« I,
e sin. u = 2 (1 - *) j tan. - v. 1 -
Substituting these values of u, and e sin. u in the equation 11 t = u —
e sin. u, we shall have the time t in a function of the anomaly v, by a series
very convergent ; but before we make this substitution, we shall observe
that (237)
n = a ~~ 2 . V m,
and since
D = a a,
we have
^ 3
1 D2
n
Hence we find
5
* V m
">
If the orbit is parabolic
a = 0
and consequently
D
1
COS. -V
V m
{tan. I + Itan.'l v}
which expression may also be got at once from (237).
The time t, the distance D and sum m of the masses of the sun and
comet, are heterogeneous quantities, to compare which, we must divide
each by the units of their species. We shall suppose therefore that the
mean distance of the sun from the Earth is the unit of distance, so that D
is expressed in parts of that distance. We may next observe that if T
VOL. II, D
50 A COMMENTARY ON [SECT. XI.
represent the time of a sidereal revolution of the Earth, setting off from
the perihelion ; we shall have in the equation
n t = u — e sin. u
u = 0 at the beginning of the revolution, and u = 2 <r at the end of it.
Hence
n T = 2 v.
But we have
_ 5
n r= a ? V m = V m,
• \/ m — . —
• « ~ «' — rp «
The value of m is not exactly the same for the Earth as for the Comet,
for in the first case it expresses the sum of the masses of the sun and
earth ; whereas in the second it implies the sum of the masses of the sun
and comet : but the masses of the Earth and Comet being much smaller
than that of the sun, we may neglect them, and suppose that m is the
same for all Planets and all Comets and that it expresses the mass of the
2 cr
sun merely. Substituting therefore for V m its value 7^- in the preced
ing expression for t ; we shall have
D*. T f 1 1 3 1
t = vvndtan-2v + stan- 2
This equation contains none but quantities comparable with each other ;
it will give t very readily when v is known ; but to obtain v by means of
t, we must resolve a Cubic Equation, which contains only one real root.
We may dispense with this resolution, by making a table of the values of
v corresponding to those of t, in a parabola of which the perihelion dis
tance is unity, or equal to the mean distance of the earth from the sun.
This table will give the time corresponding to the anomaly v, in any par
abola of which D is the perihelion distance, by multiplying by D?, the
time which corresponds to the same anomaly in the Table. We also gel
the anomaly v corresponding to the time t, by dividing t by D 2 , and
seeking in the table, the anomaly which corresponds to the quotient
arising from this division.
490. Let us now investigate the anomaly, corresponding to the time t,
in an ellipse of great excentricity.
If we neglect quantities of the order a \ and put 1 — e for a, the above
expression of t in terms of v in an ellipse, will give
D * V 2 f tan. £ v + $ tan. 3 £ v
V m ( + (1 — e) tan. 2 £ v f£ — £ tan. * £ v - J-tan. + 1 v
Then, find by the table of the motions of the comets, the anomaly cor-
BOOK L] NEWTON'S PRINCIPIA. 51
responding to the time t, in a parabola of which D is the perihelion dis
tance. Let U be this anomaly and U + x the true anomaly in an ellipse
corresponding to the same time, x being a very small angle. Then if we
substitute in the above equation U + x for v, and then transform the
second member into a series of powers of x, we shall have, neglecting the
square of x, and the product of x by 1 — e,
But by supposition
tan. £ U £1 — tan.2 £ U — | tan. 4 1 U}
U
Therefore, substituting for x its sine and substituting for sin. 4 i U its
value (1 — cos.2 1- U) 2, &c.
sin. x = Tijy (1 — e) tan. | U {4 — 3 cos. 2 % U — 6 cos. 4 \ U} .
Hence, in forming a table of logarithms of the quantity
& tan. i U [4, — 3 cos. 2 £ U — 6 cos. * \ U}
it will be sufficient to add the logarithm of 1 — e, in order to have that of
sin. x ; consequently we have the correction of the anomaly U, estimated
from the parabola, to obtain the corresponding anomaly in a very excen-
tric ellipse.
491. To find the masses of such planets as have satellites.
The equation
T = 2 ^a!i
V m
gives a very simple method of comparing the mass of a planet, having sa
tellites, with that of the sun. In fact, M representing the mass of the sun,
if (t the mass of the planet be neglected, we have
a
T _
V M
If we next consider a satellite of any planet ,«/, and call its mass p. and
mean distance from the center of (jf, h, and Tits periodic time, we shall
have
T = 2vrh^
2
_ -
M a3 T*'
This equation gives the ratio of the sum of the masses of the planet &
and its satellite to that of the sun. Neglecting therefore the mass of the
D2
52 A COMMENTARY ON [SECT. XI.
satellite, as small compared with that of the planet, or supposing their ra
tio known, we have the ratio of the mass of the planet to that of the sun.
492. To determine the Elements of Elliptical Motion.
After having exposed the General Theory of Elliptical Motion and
Method of Calculating by converging series, in the two cases of nature,
that of orbits almost circular, and the case of orbits greatly excentric, it
remains to determine the Elements of those orbits. In fact if we call V
the velocity of /* in its relative motion about M, we have
V* - dx2 + dy2 + dz*
"dTt^"
and the last of the equations (P) of No. 478, gives
To make m disappear from this expression, we shall designate by U
the velocity which P would have, if it described about M, a circle whose
radius is equal to the unity of distance. In this hypothesis, we have
e = a = i,
and consequently
U2= m.
Hence
V2 = U
This equation will give the semi-axis major a of the orbit, by means of
the primitive velocity of p and of its primitive distance from M. But a is
positive in the ellipse, and infinite in the parabola, and negative in the
hyperbola. Thus the orbit described by p is an ellipse, a parabola, or hy-
I 2
perbola, according as V is < = or > than U ^/ - . It is remarkable
that the direction of primitive motion has no influence upon the species of
conic section.
To find the excentricity of the orbit, we shall observe that if « repre
sents the angle made by the direction of the relative motion of/* with the
radius-vector, we have
dp* TT „ 9
-T-2-; = V 2 COS. 2 f.
d t2
f 2 I \
Substituting for V2 its value m | J , we have
d P2 / 2 1 \ ,
^-t — m ( — 1 cos. * £ ;
d t1 Vf a /
BOOK I.] NEWTON'S PRINCIPIA. 53
But by 480
whence we know the excentricity a e of the orbit.
To find v or the true anomaly, we have
a(l — e2)
1 -f e cos- v
a (1 — e2) — f
cos. v —
e f
This gives the position of the Perihelion. Equations (f ) of No. 480 will
then give u and by its means the instant of the Planet's passing its peri
helion.
To get the position of the orbit, referred to a fixed plane passing
through the center of M, supposed immoveable, let <p be the inclination of
the orbit to this plane, and /3 the angle which the radius f makes with the
Line of the Nodes. Let, Moreover, z be the primitive elevation of /A
above the fixed plane, supposed known. Then we
shall have, CAD being the fixed plane, A D the
line of the nodes, A B = £, &c. &c.
z = B D . sin. p r= £ sin. (3 sin. p;
so that the inclination of the oi'bit will be known
when we shall have determined ft. For this pur
pose, let X be the known angle which the primitive
direction of the relative motion of /* makes with the fixed plane ; then if
we consider the triangle formed by this direction produced to meet the
line of the nodes, by this last line and by the radius f, calling 1 the side
of the triangle opposite to 8, we have
, _ g sin. 3
" sin. (8 + i) '
Next we have
•y = sin. X.
consequently
z sin. f
tan. 8 =
sin. X — z cos. s
The elements of the Planetary Orbit being determined by these formu
las, in terms of § and z, of the velocity of the planet, and of the direction
of its motion, we can find the variation of these elements corresponding
D3
54 A COMMENTARY ON [SECT. XL
to the supposed variations in the velocity and its direction; and it will be
easy, by methods about to be explained, from hence to obtain the differ
ential variations of the Elements, due to the action of perturbing forces.
Taking the equation
V2 = U2{- —-1}.
I g a J
In the circle a = g and .*.
V = U J-
\ g
so that the velocities of the planets in different circles are reciprocally as
the squares of their radii (see Prop. IV of Princip.)
In the parabola, a = oo ,
_
the velocities in the different points of the orbit, are therefore in this case
reciprocally as the squares of the radius- vectors ; and the velocity at each
point, is to that which the body would have if it described a circle whose
radius = the radius-vector g, as V 2 : 1 (see 160)
An ellipse indefinitely diminished in breadth becomes a straight line,
and in this case V expresses the velocity of /*, supposing it to descend in
a straight line towards M. Let A* fall from rest, and its primitive dis
tance be g ; also let its velocity at the distance g' be V ; the above expres
sion will give
whence
V = U J'-^^- •
V g/
Many other results, which have already been determined after another
manner, may likewise be obtained from the above formula.
493. The equation
_dx
_
dt2
is remarkable from its giving the velocity independently of the excentricity.
It is also shown from a more general equation which subsists between the
axis-major of the orbit, the chord of the elliptic arc, the sum of the ex
treme radius-vectors, and the time of describing this arc.
To obtain this equation, we have
a(l — e2)
1 + e cos. v
BOOK I.] NEWTON'S PRINCIP1A. 55
g = a (1 — e cos. u)
3L
t == a 'J (u — e sin. a) ;
in which suppose f, v, u, and t to correspond to the first extremity of the
elliptic arc, and that p', v', u', t' belong to the other extremity ; so that we
also have
1 + e cos- v/
P' = a ( 1 — e cos. u')
t' = a 2 (u' — e sin. u').
Let now
_t' - T- u/~u - 8-
t ™" • L — J. 9 A ^— f J 9
H-£_^ = £'; g' + s = R;
then, if we take the expression oft from that oft', and observe that
sin. u' — sin. u = 2 sin. 8 cos. 8'
we shall have
T = 2 a * ?jS — e sin. jS cos. 8}.
If we add them together taking notice that
cos. u' + cos. u = 2 cos. 8. cos. 8'
we shall get
R = 2 a (1 — e cos. 8 cos. /3').
Again, if c be the chord of the elliptic arc, we have
C2=rf2 + f/2 2pf' COS. (v — v')
but the two equations
P •=. -. — \ ; P — a (1 — e cos. u)
1 4~ e cos- v
give these
cos. u — e . aVl — e2. sin. u
cos. v = a ; sn>. v =
s e
and in like manner we have
cos. u' — e , a V 1 — e" sin. u'
cos. v' = a . -f ; sin. v = , ;
whence, we get
g / cos. (v — v') = a 2 (e — cos. u) (e — cos. u') +a 2 (1 — e !) sin. u sin. u ;
and consequently
c! = 2a2(l — e2) £1 — sin. u sin. u' — cos. u cos. u'\
4- a 8 e 2 (cos. u — cos. u') * ;
D 4
56 A COMMENTARY ON [SECT. XI.
But
sin. u sin. u' + cos. u cos. u' = 2 cos. * /3 — 1
cos. u — cos. u' = 2 sin. /3 sin. /3'
.-.c2 = 4 assin.8/3(l — e2cos.2/30-
We therefore have these three equations, scil.
R = 2 a { 1 — e cos. |8 cos. 0'} ;
jT = 2 a ^ Jj3 _ e sin. j3 cos. /3'} ,
c2 = 4a2sin.2£ (1 — e*cos.2/3).
The first of them gives
a, 2 a — R
ef*/~\O /*v ~~ , __
V-WO* ^ — — _ >-»
2 a cos. p
and substituting this value of e cos. ft in the two others, we shall have
2
c2= 4a2tan.z/3|cos.2/3— (2
These two equations do not involve the excentricity e, and if in the
first we substitute for (S its value given by the second, we shall get Tina
function c, R, and a. Thus we see that the time T depends only on the
semi-axis major, the chord c and the sum R of the extreme radius-
vectors.
If we make
2 a — R + c , _ 2 a — R — c
~2lT ~^~a~~
the last of the preceding equations will give
cos. 2/3 = 22'+ V (1 — z2) . (1 — 2'2);
whence
2 j3 = cos. - ' zf — cos. - ' 2
(for cos. (A — B) = cos. A cos. B + sin. A sin. B).
Consequently
sin. (cos.-1 z') — sin. (cos. -Iz)
tan. /3 = - z + z,
we have also
2 a — R
2 + 2'=—^—.
Hence the expression of T will become, observing that if T is the du
ration of the sidereal revolution, whose mean distance from the sun is
taken for unity, we have
BOOK I.] NEWTON'S PRINCIPJA. 57
T - 2cr,
al T
T = -g — Jcos.-1 z' — cos.-' z — sin. (cos.-4 z') + sin.(cos~1z)j ... (a)
Since the same cosines may belong to many arcs, this expression is
ambiguous, and we must take care to distinguish the arcs which corre
spond to z, z'.
In the parabola, the semi-axis major is infinite, and we have
If • , A 1 /R + C\ f
cos. ~ l z — sin. (cos. — ' z ) = 3? I - ) .
6 \ a /
And making c negative we shall have the value of
cos. ~" 1 z — sin. (cos. — 1 z) ;
hence the formula (a) will give the time T employed to describe the arc
subtending the chord c, scil.
T = wzfo *< + c>*+fe + *'-') f?;
the sign — being taken, when the two extremities of the parabolic arc are
situated on the same side of the axis of the parabola.
Now T being = 365.25638 days, we have
~ = 9. 688754 days.
12 v J
The formula (a) gives the time of a body's descent in a straight line to
wards the focus, beginning from a given distance; for this, it is suffi
cient to suppose the axis-minor of the ellipse indefinitely diminished. If
we suppose, for example, that the body falls from rest at the distance 2 a
from the focus and that it is required to find the time (7") of falling to
the distance c, we shall have
R = 2 a + £, f = 2 a — c
whence
z' = _ 1, z = -
a
and the formula gives
„ a * T ( , c — a / 2 a c — c\
T = — — \9 — cos. ~ ' -- f- . / - 5 — f .
2 T I a \ a2 '
There is, however, an essential difference between elliptical motion to
wards the focus, and the motion in an ellipse whose breadth is indefinite
ly small. In the first case, the body having arrived at the focus, passes
beyond it, and again returns to the same distance at which it departed ;
but in the second case, the body having arrived at the focus immediately
returns to the point of departure. A tangential velocity at the aphelion,
58 A COMMENTARY ON [SECT. XL
however small, suffices to produce this difference which has no influence
upon the time of the body's descent to the center, nor upon the ve
locity resolved parallel to the axis-major. Hence the principles of the
7th Section of Newton give accurately the Times and Velocities, although
they do not explain all the circumstances of motion. For it is clear that
if there be absolutely no tangential velocity, the body having reached the
center offeree, will proceed beyond it to the same distance from which it
commenced its motion, and then return to the center, pass through it,
and proceed to its first point of departure, the whole being performed in
just double the time as would be required to return by moving in the in
definitely small ellipse.
494. Observations not conducting us to the circumstances of the pri
mitive motion of the heavenly bodies ; by the formulas of No. 492 we
cannot determine the elements of their orbits. It is necessary for this
end to compare together their respective positions observed at different
epochs, which is the more difficult from not observing them from the
center of their motions. Relatively to the planets, we can obtain, by
means of their oppositions and conjunctions, their Heliocentric Longitude.
This consideration, together with that of the smallness of the excentricity
and inclination of their orbits to the ecliptic, affords a very simple method
of determining their elements. But in the present state of astronomy,
the elements of these orbits need but very slight corrections ; and as the
variations of the distances of the planets from the earth are never so great
as to elude observation, we can rectify, by a great number of observations,
the elements of their orbits, and even the errors of which the observa
tions themselves are susceptible. But with regard to the Comets, this is
not feasible ; we see them only near their perihelion : if the observations
we make on their appearance prove insufficient for the determination of
their elements, we have then no means of pursuing them, even by thought,
through the immensity of space, and when after the lapse of ages, they
again approach the sun, it is impossible for us to recognise them. It be
comes therefore important to find a method of determining, by observa
tions alone during the appearance of one Comet, the elements of its orbit.
But this problem considered rigorously surpasses the powers of analysis,
and we are obliged to have recourse to approximations, in order to obtain
the first values of the elements, these being afterwards to be corrected to
any degree of accuracy which the observations permit.
If we use observations made at remote intervals, the eliminations will
lead to impracticable calculations ; we must therefore be content to con-
BOOK I.] NEWTON'S PRINCIPIA. 59
sider only near observations ; and with this restriction, the problem is abun
dantly difficult.
It appears, that instead of directly making use of observations, it is
better to get from them the data which conduct to exact and simple re
sults. Those in the present instance, which best fulfil that condition, are
the geocentric longilude and latitude of the Comet at a given instant, and
their first and second differences divided by the corresponding powers of
the element of time ; for by means of these data, we can determine rigo
rously and with ease, the elements, without having recourse to a single
integration, and by the sole consideration of the differential equations of
the orbit. This way of viewing the problem, permits us moreover, to
employ a great number of near observations, and to comprise also a con
siderable interval between the extreme observations, which will be found
of great use in diminishing the influence of such errors, as are due to ob
servations from the nebulosity by which Comets are enveloped. Let us
first present the formulas necessary to obtain the first differences, of the
longitude and latitude of any number of near observations ; and then de
termine the elements of the orbit of a Comet by means of these differences ;
and lastly expose the method which appears the simplest, of correcting
these elements by three observations made at remote intervals.
495. At a given epoch, let a be the geocentric longitude of a Comet,
and d its north geocentric latitude, the south latitudes being supposed ne
gative. If we denote by s, the number of days elapsed from this epoch,
the longitude and latitude of the Comet, after that interval, will, by using
Taylor's Theorem (481), be expressed by these two series
d ax s * /d " «\
We must determine the values of
/d as /d2 a\ /d rf\ 0
"' ld*>? (dT*)>&c" '» (ds)'&c>
by means of several observed geocentric longitudes and latitudes. To do
this most simply, consider the infinite series which expresses the geocen
tric longitude. The coefficients of the powers of s, in this series, ought to
be determined by the condition, that by it is represented each observed
longitude; we shall thus have as many equations as observations; and i(
their number is n, we shall be able to find from them, in series, the n
60 A COMMENTARY ON [SECT. XL
quantities «, f-r— ) , &c. But it ought to be observed that s being sup
posed very small, we may neglect all terms multiplied by s n, s n + l, &c.
which will reduce the infinite series to its n first terms ; which by n ob
servations we shall be able to determine. These are only approximations,
and their accuracy will depend upon the smallness of the terms which are
omitted. They will be more exact in proportion as s is more diminutive,
and as we employ a greater number of observations. The theory of inter
polations is used therefore To find a rational and integer function qfs such,
that in substituting therein for s the number of days 'which correspond to each
observation, it shall become the observed longitude.
Let (3, /3', f3"f &c. be the observed longitudes of the comet, and by
i, i', i", &c. the corresponding numbers of days from the given epoch, the
numbers of the days prior to the given epoch being supposed negative.
If we make
R' R R" R> R'" R
p — a B • — " - — 3 fi' "
'' " • — ° P ) •// •/ — OP
>l!f-tf> •-> '
1" — 1 1'
y — d2^
; &c.
i'" — i
&c.;
the required functions will be
for it is easy to perceive that if we make successively s = i, s = i', s = i", &c.
it will change itself into /3, /3', /3/x, &c.
Again, if we compare the preceding function with this
we shall have by equating coefficients of homogeneous terms.
i'. 6 2 /3 — i . i'. \"
— &c.
The higher differences of a will be useless. The coefficients of these
expressions are alternately positive and negative ; the coefficient of d r 13
is, disregarding the sign, the product of r and r together of r quantities
i, i', . . . . i (r-1! in the value of « ; it is the sum of the products of the
BOOK L] NEWTON'S PRINCIPIA.
same quantities, r — 1 together in the value of (-r
of the products of these quantities r — 2, together in the value of
^d s2>
If 7, 7', 7", &c. be the observed geocentric latitudes, we shall have the
values of d, (-r— ) , (-1 — 2) > &c. by changing in the preceding expressions
for a (-p) 5 ( i — ") 5 &c. the quantities /3, (3', /3" into 7, /, 7".
These expressions are the more exact, the greater the number of ob
servations and the smaller the intervals between them. We might,
therefore, employ all the near observations made at a given epoch, pro
vided they were accurate; but the errors of which they are always sus
ceptible will conduct to imperfect results. So that, in order to lessen the
influence of these errors, we must augment the interval between the ex
treme observations, employing in the investigation a greater number of
them. In this way with five observations we may include an interval of
thirty-five or forty degrees, which would give us very near approximations
to the geocentric longitude and latitude, and to their first and second
differences.
If the epoch selected were such, that there were an equal number of
observations before and after it, so that each successive longitude may
have a corresponding one which succeeds the epoch. This condition will
give values still more correct of a, f-t— 'J and ( ,- — -) j an(^ it easily appears
that new observations taken at equal distances from either side of the epoch,
would only add to these values, quantities which, with regard to their last
i g
terms, would be as s 2 ( -. 2 j to «. This symmetrical arrangement takes
place, when all the observations being equidistant, we fix the epoch at
the middle of the interval which they comprise. It is therefore advanta
geous to employ observations of this kind.
In general, it will be advantageous to fix the epoch near the middle of
this interval ; because the number of days included between the extreme
observations being less considerable, the approximations will be more con
vergent. We can simplify the calculus still more by fixing the epoch at
the instant of one of the observations ; which gives immediately the values
of «, and 6.
62 A COMMENTARY ON [SECT. XI
When we shall have determined as above the values of
d«\ /d2«\ /d S\ i /d2
«\ / \ i / \
T2)' GB)I and (dp)
we shall then obtain as follows the first and second differences of a, and fl
divided by the corresponding powers of the elements of time. If we neg
lect the masses of the planets and comets, that of the sun being the unit
of mass ; if, moreover, we take the distance of the sun from the earth for
the unit of distance ; the mean motion of the earth round the sun will
be the measure of the time t. Let therefore X be the number of se
conds which the earth describes in a day, by reason of its mean sidereal
motion ; the time t corresponding to the number of days will be X s ; we
shall, therefore, have
(d «\ 1 /d «\
d~~t/ " T \dl)
(d2a\ 1 /d2a\
d"tV ~ X~Hd sV*
Observations give by the Logarithmic Tables,
log. X = 4. 0394622
and also
log. X 2 = log. X + log. -g
R bein the radius of the circle reduced to seconds ; whence
log. Xs = 2.2750444;
J J 2
.-. if we reduce to seconds, the values of (-p) 5 and of (-T — 2) , we shall
1
have the logarithms of ( ,-") , and of (^-^) by taking from the logarithms
*C1 t' »Cl I /
of these values the logarithms of 4. 039422, and 2. 2750444. In like
manner we get the logarithms of ( r-V (T .4) , after subtracting the
same logarithms, from the logarithms of their values reduced to seconds.
On the accuracy of the values of
d
depends that of the following results ; and since their formation is very
simple, we must select and multiply observations so as to obtain them with
the greatest exactness possible. We shall determine presently, by means
of these values, the elements of the orbit of a Comet, and to generalize
these results, we shall
BOOK I.] NEWTON'S PRINCIPIA. 63
496. Investigate the motion of a system of bodies sollicited by any forces
whatever.
Let x, y, z be the rectangular coordinates of the first body ; x', y', z'
tliose of the second body, and so on. Also let the first body be sollicited
parallel to the axes of x, y, z by the forces X, Y, Z, which we shall sup
pose tend to diminish these variables. In like manner suppose the second
body sollicited parallel to the same axes by the forces X', Y', Z', and so
on. The motions of all the bodies will be given by differential equations
of the second order
&c. = &c.
If the number of the bodies is n, that of the equations will be 3 n ; and
their finite integrals will contain 6 n arbitrary constants, which will be the
elements of the orbits of the different bodies.
To determine these elements by observations, we shall transform the
coordinates of each body into others whose origin is at the place of the
observer. Supposing, therefore, a plane to pass through the eye of the
observer, and of which the situation is always parallel to itself, whilst the
observer moves along a given curve, call r, r' r", &c. the distances of
the observer from the different bodies, projected upon the plane ;
«, a', a", &c. the apparent longitudes of the bodies, referred to the same
plane, and 6, ff, 0", &c. their apparent latitudes. The variables x, y, z
will be given in terms of r, «, 0, and of the coordinates of the observer.
In like manner, x', y', z' will be given in functions of r7, a', ff, and of the
coordinates of the observer, and so on. Moreover, if we suppose that the
forces X, Y, Z ; X7, Y', Z', &c. are due to the reciprocal action of the
bodies of the system, and independent of attractions ; they will be given in
functions of r, r', r", &c. ; a, a', a", &c. ; 6, 6', 6", &c. and of known quan
tities. The preceding differential equations will thus involve these new
variables and their first and second differences. But observations make
known, for a given instant, the values of
/d ax /d*«\ . /d 0\ /d2 6\ , /da'\
*> (ai)' (arO' Men)' (dT*); "'Car)' &c-
There will hence of the unknown quantities only remain r, r7, r", &c.
and their first and second differences. These unknowns are in number
3 n, and since we have 3 n differential equations, we can determine them.
64 A COMMENTARY ON [SECT. XI.
At the same time we shall have the advantage of presenting the first and
second differences of r, r', r", &c. under a linear form.
The quantities «, 6, r, «', ^, r7, &c. and their first differences divided by
d t, being known ; we shall have, for any given instant, the values of
x, y, z, x', y', z', &c. and of their first differences divided by d t. If we
substitute these values in the 3 n finite integrals of the preceding equa
tions, and in the first differences of these integrals ; we shall have 6 n
equations, by means of which we shall be able to determine the 6 n arbi
trary constants of the integrals, or the elements of the orbits of the dif
ferent bodies.
497. To apply this method to the motion of the Comets,
We first observe that the principal force which actuates them is the
attraction of the sun ; compared with which all other forces may be ne
glected. If, however, the Comet should approach one of the greater
planets so as to experience a sensible perturbation, the preceding method
will still make known its velocity and distance from the earth ; but this
case happening but very seldom, in the following researches, we shall ab
stain from noticing any other than the action of the sun.
If the sun's mass be the unit, and its mean distance from the earth the
unit of distance; if, moreover, we fix the origin of the coordinates
x, y, z of a Comet, whose radius-vector is g ; the equations (0) of No. 475
will become, neglecting the mass of the Comet,
o _ ,
2
(k)
dt2
Let the plane of x, y be the plane of the ecliptic. Also let the axis of
x be the line drawn from the center of the sun to the first point of aries,
at a given epoch ; the axis of y the line drawn from the center of the sun
to the first point of cancer, at the same epoch ; and finally the positive
values of z be on the same side as the north pole of the ecliptic. Next
call x', y7 the coordinates of the earth and R its radius-vector. This be
ing supposed, transfer the coordinates x, y, z to others relative to the
observer ; and to do this let a be the geocentric longitude, and r its dis
tance from the center of the earth projected upon the ecliptic ; then we
shall have
x = x' -f- r cos. «; y = y + r sin. a; z = r tan. 6.
BOOK I.] NEWTON'S PRINCIPIA. 66
If we multiply the first of equations (k) by sin. a, and take from the re
sult tlve second multiplied by cos. a, we shall have
d2x d 2 y x sin. a — y cos. a
whence we derive, by substituting for x, y their values given above,
d 2 x' d ° y' x' sin. a — y cos. a
= s..Trr-. -
d r\ /da
The earth being retained in its orbit like a comet, by the attraction of
the sun, we have
dlx , *_ n _ dV , jr
dt2 + RS} ' ~dTt2 + RS;
which give
We shall, therefore, have
d 2 x' d 2 V y' cos. a — x' sill, a
sin. a — — - — cos. a . .-. v = =• =^
d t z dt* R 3
n / • da /d2a
-
Let A be the longitude of the earth seen from the sun ; we shall have
x' = R cos. A ; y' = R sin. A ;
therefore
y' cos. a — x' sin. a = R sin. (A — a) ;
and the preceding equation will give
/d»«x
/drx Rsin.(A — a) M 1) ' \d tV
Vdt/= /dUx '(R3""^/- -TJ-
-
Now let us seek a second expression for (j~\ . For this purpose we
will multiply the first of equations (k) by tan. & . cos. «, the second by
tan. 6 sin. a, and take the third equation from the sum of these two pro
ducts ; we shall thence obtain
sn.
\- tan 6 x cos- a + y sin*
3
_ _ _
S3 ~dt2 g3'
This equation will become by substitution for x, y, z
./ /d2x/ , x\ , /d2y' y\ . \
0 = tan. l( (^ + ^) cos. « + (^ + £) sin. «}
VOL. TI. K
66
A COMMENTARY ON [SECT. XI.
But
,.- = co,
= R cos . ( A — « ) — — g-
Therefore,
R sin. 6 cos. 0 cos. (A — «) / 1 1 ) /2\
+ - -TdV -tr"R3/'
' Vd J
If we take this value of (^) from the first and suppose
sin. tf cos. tf cos. (A — ) + ( ) sin. (A - «)
we shall have
The projected distance r of the comet from the earth, being always po
sitive, this equation shows that the distance s of the comet from the sun,
is less or greater than the distance R of the sun from the earth, according
as (i! is positive or negative; the two distances are equal if (if = 0.
By inspection alone of a celestial globe, we can determine the sign of
// ; and consequently whether the comet is nearer to or farther from the
Earth. For that purpose imagine a great circle which passes through
two Geocentric positions of the Comet infinitely near to one another.
Let 7 be the inclination of this circle to the ecliptic, and X the longitude
of its ascending node ; we shall have
tan. 7 sin. (a — X) = tan. 6 ;
\vricncc
d 6 sin. (a — X) = a a sin. 6 cos. 6 cos. (a — X).
BOOK I.] NEWTON'S PRINCIPIA. 67
Differentiating, we have, also
/dav/d*0\ /d<K/d2a\( 0/dax/d0\8
0 = (di) (jtO-VD Grrv + 2vdi) (di) lan- '
I 3
+ ( v-r)
sn. cos.
d20, being the value of d2 6, which would take place, if the apparent mo
tion of the Comet continued in the great circle. The value of y! thus be
comes, by substituting for d 6 its value
d a sin. 6 cos. 6 cos. (a • — X)
sin. (a — X)
sin. 0 cos. 0 sin. (A — X)
The function . ' V - .- is constantly positive : the value of IL is there-
sin. 6 cos. 6
c • • • i- /d2 6\ /d'^Ai.
k>re positive or negative, according asf-j— ^J — (-p-pjhas the same or
a different sign from that of sin. (A — X). But A — X is equal to two
right angles plus the distance of the sun from the ascending node of the
great circle. Whence it is easy to conclude that fjf will be positive or
negative, according as in a third geocentric position of the comet, inde
finitely near to the two first, the comet departs from the great circle on
the same or the opposite side on which is the sun. Conceive, therefore,
that we make a great circle of the sphere pass through the two geocentric
positions of the comet ; then according as, in a third consecutive geocen
tric position, the comet departs from this great circle, on the same side as
the sun or on the opposite one, it will be nearer to or farther from the
sun than the Earth. If it continues to appear in this great circle, it will
be equally distant from both ; so that the different deflections of its ap
parent path points out to us the variations of its distance from the sun.
To eliminate § from equation (3), and to reduce this equation so as to
contain no other than the unknown r, we observe that g2 = xz + y2 -f- z*
in substituting for x, y, z, their values in terms of
r, a, and 0 ;
and we have
Sa- = x'2 + y/2+ 2r£x'cos. a + y' sin. a] + — ^ J
but we have
x' — R cos. A, y' = R sin. A ;
c^ + 2 R r cos- <A - a) + Il';
E2
68 A COMMENTARY ON [SECT. XL
But
x' = R cos. A ; y' = R sin. A
.-. P 2 = -^r, + 2 R r cos. (A — «) + tt 2-
cos. 2 6
If we square the two members of equation (3) put under this form
e*{p' R2r + 1}= R3
we shall get, by substituting for g 2,
/ -^-— -f 2 R r cos. (A — «) + R2 j .{(*' R2 r + l}'= Rc . . . (4)
\ cos. 2 6 )
an equation in which the only unknown quantity is r, and which will rise
to the seventh degree, because a term of the first member being equal to
R 6, the whole equation is divisible by r. Having thence determined r,
we shall have (-.— ) by means of equations (1) and (2). Substituting, for
example, in equation (1), for — 3 — R-, its value -~ , given by equation
(3) ; we shall have
The equation (4) is often susceptible of many real and positive roots ;
reducing it and dividing by r, its last term will be
2 R5 cos.6 W R3 + 3 cos. (A — a)}.
Hence the equation in r being of the seventh degree or of an odd de
gree, it will have at least two real positive roots if [if R 3 + 3 cos. (A — a)
is positive; for it ought always, by the nature of the problem, to have
one positive root, and it cannot then have an odd number of positive
roots. Each real and positive value of r gives a different conic section,
for the orbit of the comet ; we shall, therefore, have as many curves
which satisfy three near observations, as r has real and positive values ;
and to determine the true orbit of the comet, we must have recourse to a
new observation.
498. The value of r, derived from equation (4) would be rigorously
exact, if
were exactly known ; but these quantities are only approximate. In fact,
by the method above exposed, we can approximate more and more, mere
ly by making use of a great number of observations, which presents the
advantage of considering intervals sufficiently great, and of making the
errors arising from observations compensate one another. But this
BOOK I.] NEWTON'S PUINCIPIA. 69
method has the analytical inconvenience of employing more than three
observations, in a problem where three are sufficient. This may be
obviated, and the solution rendered as approximate as can be wished by
three observations only, after the following manner.
Let a and 6, representing the geocentric longitude and latitude of the
intermediate ; if we substitute in the equations (k) of the preceding
No. instead of x, y, z their values x' + r cos. a ; y' + r sin. a ; and
r tan. 6; they will give (_.— 2V ( i 1 2) anc^ ("rT2) m ^unc^ons °f r> "> and
0, of their first differences and known quantities. If we differentiate these,
we shall havef-. — j} , (-^ — 5} and (-T — 3} in terms of r, a, 6, and of their
first and second differences. Hence by equation (2) of 497 we may eli
minate the second difference of r by means of its value and its first differ
ence. Continuing to differentiate successively the values of (-r— ) > (--, — 3) •>
and eliminating the differences of a, and of 0 superior to second differences,
and all the differences of r, we shall have the values of
d
' &Ct in terms ot
d d /d2 tfv
this being supposed, let
«/> a, a',
be the three geocentric observed longitudes of the Comet; 0/3 0, tf its
three corresponding geocentric latitudes; let i be the number of days
which separate the first from the second observation, and i' the interval
between the second and third observation ; lastly let X be the arc which
the earth describes in a day, by its mean sidereal motion ; then by (481)
we have
. . /d «\ , i 2. X 2 /d 2 «\ i 3 X 3 fd 3 «\
"< = «-•• Hen) + TTW- naCfw + &c-;
, , ., , /da\ i/2. X2/d2ax i/3. X3/d3«\
= « + 1' . x (d-t) + L 2 (^ + 1^3 (a T3) + &c. ,
/v .x2/2^
Cdl) + 172- (dl -0
2
70 A COMMENTARY ON [Stcx. XI.
If we substitute in these series for
their values obtained above, we shall have four equations between the
five unknown quantities
These equations will be the more exact in proportion as we consider a
greater number of terms in the series. We shall thus have
/«\ / a\ / \ / \
\d~t)> \2TtV! vTt/' VdT2,)
in terms of r and known quantities; and substituting in equation (4) of
the preceding No. it will contain the unknown r only. As to the rest,
this method, which shows how to approximate to r by employing three
observations only, would require in practice, laborious calculations, and
it is a more exact and simple process to consider a greater number of ob
servations by the method of No, 495.
499. When the values of r and f-i~J shall be determined, we shall have
those of
/d x\ /d y\ , /d z\
x'>''McTt)' (dt)and(dl)'
by means of the equations
x = R cos. A + r cos. a
y = R sin. A + r sin. «
z = r tan. 6
and of their differentials divided by d t, viz.
dx\ /d R\ T, /d A\ . /d r\
s- A - Rs'n- A + cos-
/ r\
- A + (ai)
v\ /d R\ . ,, /d A\
t ) = ( dnr)sm- A + R( d v)cos- A
—
d t
The values of ((\A) and of (A—) are given by the Theory of the
motion of the Eai'th :
To facilitate the investigation let E be the excentricity of the earth's
BOOK I.] NEWTON'S PRINCIPIA. 71
orbit, and H the longitude of its perihelion; then by the nature of
elliptical motion we have
/dAx V(i-E2). _ 1-E2
VdT/ ~U*~ • 1 + Ecos. (A — H)*
These two equations give
/d Rx E sin. (A — H)
Idt/ : ~V (1 — E2)
Let R' be the radius- vector of the earth corresponding to the longitude
A of this planet augmented by a right angle ; we shall have
___ _
' 1 — E sin. ( A — H) '
whence is derived
T-, • /A Tjv R' — 1 +
E sin. ( A — H) = - — -:
/d Rx R/ + E «_-- 1
' \dt) ~ R' — V (1 — E2)'
If we neglect the square of the excentricity of the earth's orbit, which is
very small, we shall have
/d A\ _ 1 /d Rx ,
ITF)-R-«; (dr) = R -
the preceding values of (T~T) and f -p2" Will hence become
d xx _ sin. A , /d r\ /d«\ •
)cos-A- ~ir+ (di)cos-a-r(di)sln-a;
- cos- A /d
fy\ /-of i\ -
(df) =(R- ^ Sm'
R, R', and A being given immediately by the tables of the sun, the esti
mate of the six quantities x, y, z, ( j- ^) ' (d~?) ' (d?) wil1 be
when r and - shall be known. Hence we derive the elements of the
orbit of the comet after this mode.
The indefinitely small sector, which the projection of the radius- vector
and the comet upon the plane of the ecliptic describes during 'the element
of time d t, is - —21_XJ - • and it is evident that this sector is posi
tive or negative, according as the motion of the comet is direct or retro
grade. Thus in forming the quantity x (jl) _ y (1~), it will indicate
by its sign, the direction of the motion of the comet.
E 4,
72 A COMMENTARY ON [SECT. XI.
To determine the position of the orbit, call <p its inclination to the
ecliptic, and I the longitude of the node, which would be ascending if the
motion of the comet were direct or progressive. We shall have
z = y cos. I tan. <p — x sin. I tan. <p
These two equations give
tan. I =
tan. =
Wherein since <p ought always to be positive and less than a right
angle, the sign of sin. I is known. But the tangent of I and the sign of
its sine being determined, the angle I is found completely. This angle
is the longitude of the ascending node of the orbit, if the motion is pro
gressive; but to this we must add two right angles, in order to get the
longitude of the node when the motion is retrograde. It would be more
simple to consider only progressive motions, by making vary p, the in
clination of the orbits, from zero to two right angles ; for it is evident that
then the retrograde motions correspond to an inclination greater than a
right angle.
In this case, tan. <p has the same sign as x ( j^-) — y (-i — ) > which will
determine sin. I, and consequently the angle I, which always expresses
the longitude of the ascending node.
If a, a e be the semi-axis major and the excentricity of the orbit, we
have (by 492) in making m = 1,
The first of these equations gives the semi-axis major, and the second
the excentricity. The sign of the function x (j^) + ? (j~D + z (dl)
shows whether the comet has already passed its perihelion ; for it ap
proaches if this function is negative; and in the contrary case, the comet
recedes from that point.
BOOK I.] NEWTON'S PRINCIPIA. 73
Let T be the interval of time comprised between the epoch and pas
sage of the comet over the perihelion; the two first of equations (f) (480)
_5
will give, observing that m being supposed unity we have n = a 2 ,
£ = a (1 — e cos. u)
5
T = a 2 (u — e cos. u).
The first of these equations gives the angle u, and the second T. This
time added to or subtracted from the epoch, according as the comet ap
proaches or leaves its perihelion, will give the instant of its passage over
this point. The values of x, y, determine the angle which the projection
of the radius-vector £ makes with the axis of x ; and since we know the an
gle I, formed by this axis and by the line of the nodes, we shall have the
angle which this last line forms with the projection of g ; whence we derive by
means of the inclination p of the orbit, the angle formed by the line of the
nodes and the radius f. But the angle u being known, we shall have by
means of the third of the equations (f), the angle v which this radius forms
with the line of the apsides ; we shall therefore have the angle comprised
between the two lines of the apsides and of the nodes, and consequently,
the position of the perihelion. All the elements of the orbit will thus be
determined.
500. These elements are given, by the preceding investigations, in terms
of r, (-1-7) and known quantities ; and since (-,- ) is given in terms of r
by No. 497, the elements of the orbit will be functions of r and known
quantities. If one of them were given, we should have a new equation,
by means of which we might determine r ; this equation would have a
common divisor with equation (4) of No. 497; and seeking this di
visor by the ordinary methods, we shall obtain an equation of the first
degree in terms of r ; we should have, moreover, an equation of condition
between the data of the observations, and this equation would be that
which ought to subsist, in order that the given element may belong to the
orbit of the comet.
Let us apply this consideration to the case of nature. First suppose
that the orbits of the comets are ellipses of great excentricity, and are
nearly parabolas, in the parts of their orbits in which these stars are
visible. We may therefore without sensible error suppose a = <x>, and
consequently - = 0; the expression for - of the preceding No. will there
fore give
74 A COMMENTARY ON [SECT. XI.
2 dx2 + dy2 + dz2
: 7 u dt2
If we then substitute for f-r—\ IT-M and (-5—) their values found in
vd tJ \d t/ \d t/
the same No., we shall have after all the reductions and neglecting the
square of R' — 1,
•- (So* ••(
cos.
2,
Substituting in this equation for (-5— ) its value
J/d2«\ , xl
• i ( -5 — 5- ) + /* sin. (A — a) f ,
ax I \d t2/ ^ 'J '
found in No. 497, and then making
./d«\z T, . /d«\ 4 . f /d2 a\ , . . /A N) "
Hal) • B = Hai) + 1 (err*) + " sin- <A - "U
C /d «\ /d ^\ ^ 2
J tan. 6. (-, — -a} + ij,' tan. 6 sin. (A — «) — — 4
v. \ct t / cos. 9 J
and
C = d t -JL /5El^ZI^__(R'__i)cos. (A — «)l
/d«\ ( K J
we shall have
0 = Br2+ Cr + ^i— ~
and consequently
r*+ Cr + -2=: 4.
This equation rising only to the sixth degree, is in that respect, more
BOOK I.] NEWTON'S PRINCIPIA. 75
simple than equation (4) of No. (497) ; but it belongs to the parabola
alone, whereas the equation (4) equally regards every species of conic
section.
501. We perceive by the foregoing investigation, that the determina
tion of the parabolic orbits of the comets, leads to more equations than
unknown quantities ; and that, therefore, in combining these equations in
different ways, we can form as many different methods of calculating the
orbits. Let us examine those which appear to give the most exact re
sults, or which seem least susceptible of the errors of observations.
It is principally upon the values of the second differences f-r — ^] and
/d 2 d\
( -j — ; ), that these errors have a sensible influence. In fact, to determine
\d. t~s
them, we must take the finite differences of the geocentric longitudes and
latitudes of the comet, observed during a short interval of time. But
these differences being less than the first differences, the errors of obser
vations are a greater aliquot part of them ; besides this, the formulas of
No. 495 which determine, by the comparison of observations, the values
c , /d«\ /d 0\ /d2 «\ , /d 2 d\ . . , . . .
ot «, 6, Ijriji (TT)' ITTTJ anc^ VTTV Swe greater precision the
four first of these quantities than the two last. It is, therefore, desirable
to rest as little as possible upon the second differences of « and 6; and
since we cannot reject both of them together, the method which employs
the greater, ought to give the more accurate results. This being granted
let us resume the equations found in Nos. 497, &c.
'* = dr2i + 2Rrc°St(A~a)"f RJ;
£x R sin. (A — «) fj_ H_
J : /cUv IK> " g3/"
2 Iff*!
Y sin.* cos. A
J (
~7d ^ i
vai>
R sin. 5 cos. ^ cos. (A — a)
76 A COMMENTARY ON [SECT. XL
-! co, _
+ 2 , (**) {(R' - 1) sin. (A - „) + "MA— )}
i
h If5
/(I 2 $\
If we wish to reject (^ — 5) , we consider only the first, second and fourth
of those equations. Eliminating (-7— ) from the last by means of the
second, we shall form an equation which cleared of fractions, will contain
a term multiplied by g 6 r2, and other terms affected with even and odd
powers of r and g. If we put into one side of the equation all the terms
affected with even powers of g, and into the other all those which involve
its odd powers, and square both sides, in order to have none but even
powers of f, the term multiplied by § 6 r 2 will produce one multiplied by
g12 r4. Substituting, therefore, instead of g2, its value given by the first
of equations (L), we shall have a final equation of the sixteenth degree in
r. But instead of forming this equation in order afterwards to resolve it,
it will be more simple to satisfy by trial the three preceding ones.
If we wish to reject ( -, — 5), we must consider the first, third and fourth
of equations (L). These three equations conduct us also to a final equa
tion of the sixteenth degree in r ; and we can easily satisfy by trial.
The two preceding methods appear to be the most exact, which we can
employ in the determination of the parabolic orbits of the comets. It is
at the same time necessary to have recourse to them, if the motion of the
comet in longitude or latitude is insensible, or too small for the errors of
observations sensibly to alter its second difference. In this case, we must
reject that of the equations (L), which contains this difference. But al
though in these methods, we employ only three equations, yet the fourth
is useful to determine amongst all the real and positive values of r, which
satisfy the system of three equations, that which ought to be selected.
502. The elements of the orbit of a comet, determined by the above
process, would be exact, if the values of a, 6 and their first and second
differences, were rigorous ; for we have regarded, after a very simple
manner, the excentricity of the terrestrial orbit, by means of the radius-
vector R' of the earth, corresponding to its true anomaly + & right an
gle ; we are therefore permitted only to neglect the square of this excen-
BOOK I.] NEWTON'S PIUNCIP1A. 77
tricity, as too small a fraction to produce by its omission a sensible influ
ence upon the results. But 0, a and their differences, are always suscep
tible of any degree of inaccuracy, both because of the errors of observa
tions, and because these differences are only obtained approximately. It
is therefore necessary to correct the elements, by means of three distant
observations, which can be done in many ways ; for if we know nearly,
two quantities relative to the motion of a comet, such that the radius-vec
tor corresponding to two observations, or the position of the node, and
' the inclination of the orbit ; calculating the observations, first with these
quantities and afterwards with others differing but little from them, the
law of the differences between the results, will easily show the necessary
corrections. But amongst the combinations taken two and two, of the
quantities relative to the motion of comets, there is one which ought to
produce greatest simplicity, and which for that reason should be selected.
It is of importance, in fact, in a problem so intricate, and complicated, to
spare the calculator all superfluous operations. The two elements which
appear to present this advantage, are the perihelion distance, and the
instant when the comet passes this point. They are not only easy to be
derived from the values of r and -p- ) ; but it is very easy to correct them
by observations, without being obliged for every variation which they
undergo, to determine the other corresponding elements of the orbit.
Resuming the equation found in No. 492
a (1 — e2) is the semi-parameter of the conic section of which a is the
semi axis-major, and a e the excentricity. In the parabola, where a is
infinite, and e equal to unity, a (1 — e 2) is double the perihelion dis
tance : let D be this distance : the preceding equation becomes relatively
to this curve
pde £ d P 2 r2
-.— is equal to-^—5-; in substituting for e2its value r-:+2RrX
cl t at2 cos. 2 0
COS.
R>
cos. (A — a) + R2, and for (-3-7] and (^rr)1 their values found in
No. 499, we shall have
d t cos. 2 &
78 A COMMENTARY ON [SECT. XI.
+ r{(R'_ 1) cos. (A-«)-.gin-<^-g)}
+ r R ~ sin. (A — «) + R (R' — 1).
Let P represent this quantity ; if it is negative, the radius-vector de
creases, and consequently, the comet tends towards its perihelion. But
it goes off into the distance, if P is negative. We have then
D = S-IP*;
the angular distance v of the comet from its perihelion, will be determined
from the polar equation to the parabola,
cor-2!v = 7;
and finally we shall have the time employed to describe the angle v, by
the table of the motion of the comets. This time added to or subtracted
from that of the epoch, according as P is negative or positive, will give
the instant when the comet passes its perihelion.
503. Recapitulating these different results, we shall have the following
method to determine the parabolic orbits of the comets.
General method of determining the orbits of the comets.
This method will be divided into two parts ; in the first, we shall give
the means of obtaining approximately, the perihelion distance of the comet
and the instant of its passage over the perihelion ; in the second, we shall
determine all the elements of the orbit on the supposition that the former
are known.
Approximate determination of the Perihelion distance of the comet, and
the instant of its passage over the perihelion,
We shall select three, four, five, &c. observations of the comet
equally distant from one another as nearly as possible ; with four obser
vations we shall be able to consider an interval of 30° ; with five, an in
terval of 36°, or 40° and so on for the rest ; but to diminish the in
fluence of their errors, the interval comprised between the observations
must be greater, in proportion as their number is greater. This being
supposed,
Let /3, /3', (3", &c. be the successive geocentric longitudes of the comet,
7, /, /' the corresponding latitudes, these latitudes being supposed positive
or negative according as they are north or south. We shall divide the dif
ference 13' — |8, by the number of days between the first and second ob
servation ; we shall divide in like manner the difference ft" — P by the
BOOK I.] NEWTON'S PRINCIPIA. 79
number of days between the second and third observation ; and so on.
Let 3 8, d B', d B", &c. be these quotients.
We next divide the diffei-ence 88' — SB by the number of days be
tween the first observation and the third ; we divide, in like manner, the
difference 8 B" — d $' by the number of days between the second and
fourth observations ; similarly we divide the difference 8 B"' — 8 B" by the
number of days between the third and fifth observation, and so on. Let
8 2 18, 8 2 6', & 2 /3", &c. denote these quotients.
Again, we divide the difference B z B' — 8 2 B by the number of days
which separate the first observation from the fourth ; we divide in like
manner 8 2 B" — 8 2 B' by the number of days between the second obser
vation and the fifth, and so on. Make 8 3 8, 83 8', &c. these quotients.
Thus proceeding, we shall arrive at 8 n - l 89 n being the number of obser
vations employed.
This being done, we proceed to take as near as may be a mean epoch
between the instants of the two extreme observations, and calling i, i', i",
&c. the number of days, distant from each observation, i, i', i", Sec. ought
to be supposed negative for the observations made prior to this epoch ;
the longitude of the comet, after a small number z of days reckoned from
the Epoch will be expressed by the following formula :
j3 _ i a 8 + i i' d 2 B — i i' i" 8 3 B + &c.
\ +ZJ3 8—(i + i')8 Z8+ (i i' + i i"+i' i")3 3B— (i i' i"+i i' i'" + i i" i'"+. . (p)
)i'i"i"'
V. 2
The coefficients of — 8 8, + 8 z B, — 8 3 8, &c. in the part independent
of z are 1st the numbers i and i', secondly the sum of the products two
and two of the three numbers i, i', \" ; thirdly the sum of the products
three and three, of the four numbers i, i', i", i"', &c.
The coefficients of — 83 B, + 8 4 8, — 8 5 8, &c. in the part multiplied
by z 2, are first, the sum of the three numbers i, i', i7' ; secondly of the
products two and two of the four numbers i, i', i'', i'"; thirdly the sum of
the products three and three of the five numbers i, i', i", i"', i"", &c.
Instead of forming these products, it is as simple to develope the func
tion B + (z — i) 6/3 + (z — i) (z — i') 62/3+ (z — i) (z— i') (z — i")
X 6 3 8 -f- &c. rejecting the powers of z superior to the square. This
gives the preceding formula.
If we operate in a similar manner upon the observed geocentric lati
tudes of the comet ; its geocentric latitude, after the number z of days
from the epoch, will be expressed by the formula (p) in changing 8 into
7. Call (q) the equation (p) thus altered. This being done,
80 A COMMENTARY ON [SECT. XL
a will be the part independent of z in the formula (p) ; and 6 that in the
formula (q).
Reducing into seconds the coefficient of z in the formula (p), and
taking from the tabular logarithm of this number of seconds, the logarithm
4,0394622, we shall have the logarithm of a number which we shall de
note by a.
Reducing into seconds the coefficients of z2 in the same formula, and tak-
ing from the logarithm of this number of seconds, the logarithm 1.9740144,
we shall have the logarithm of a number, which we shall denote by b.
Reducing in like manner into seconds the coefficients of z and z 2 in
the formula (q) and taking away respectively from the logarithms of these
numbers of seconds, the logarithms, 4,0394622 and 1,9740144, we shall
have the logarithms of two numbers, which we shall name h and 1.
Upon the accuracy of the values of a, b, h, 1, depends that of the
method; and since their formation is very simple, we must select and
multiply observations, so as to obtain them with all the exactness which
the observations will admit of. It is perceptible that these values are only
/dax /d2ax /d 6\ /d2 6\
the quantities (^J > VdT2/ ' \d~t/ ' VdT2/ ' wmch we have exPress'
ed more simply by the above letters.
If the number of observations is odd, we can fix the Epoch at the
instant of the mean observation; which will dispense with calculating the
parts independent of z in the two preceding formulas ; for it is evident,
that then these parts are respectively equal to the longitude and latitude
of the mean observation.
Having thus determined the values of a, a, b, 8, h, and 1, we shall de
termine the longitude of the sun, at the instant we have selected for the
epoch, R the corresponding distance of the Earth from the sun, and R'
the distance which answers to E augmented by a right angle. We shall
have the following equations
(1)
(2)
( '
§!
y
y
X2
cos. 2 6
^ sin. (E — a
x cos. (E — a) + R2
) r i i i bx
2a
= — x -j h tan. d
( g3 R3j 2 a
1 l a2 sin. 6 . cos.
iVi
1 2h ' 2h
if
R sin. ^ cos. ^ f ^ ^ ' ' (3;
-tr~ cos-
BOOK I.] NEWTON'S PRINCIPIA. 81
— (IT — 1) cos. (E — a)} — 2 a x -[(R/ — 1) sin. (E — a) +
'
To derive from these equations the values of the unknown quantities
Xj y> & we must consider, signs being neglected, whether b is greater or
less than 1. In the first case we shall make use of equation (1), (2), and
(4). We shall form a first hypothesis for x, supposing it for instance
equal to unity; and we then derive by means of equations (1), (2), the
values of § and of y. Next we substitute these values in the equation (4) ;
and if the result is 0, this will be a proof that the value of x has been
rightly chosen. But if it be negative we must augment the value of x,
and diminish it if the contrary. We shall thus obtain, by means of a
small number of trials the values of x, y and g. But since these unknown
quantities may be susceptible of many real and positive values, we must
seek that which satisfies exactly or nearly so the equation (3).
In the second case, that is to say, if 1 be greater than b, we shall use
the equations (1), (3), (4), and then equation (2) will give the verifi
cation.
Having thus the values of x, y, g, we shall have the quantity
p = & + h x tan> 6}~ R y cos- (E K)
+ x = _(R'_ 1) cos. (E— *) — Rax rin (E-«)
+ R.(R/— 1).
The Perihelion distance D of the comet will be
D = s-lp*;
the cosine of its anomaly v will be given by the equation
„ 1 D
cos^-v = -;
and hence we obtain, by the table of the motion of the comets, the time
employed to describe the angle v. To obtain the instant when the comet
passes the perihelion, we must add this time to, or subtract it from the
epoch according as P is negative or positive. For in the first case the
comet approaches, and in the second recedes from, the perihelion.
Having thus nearly obtained the perihelion distance of the comet, and
the instant of its passage over the perihelion ; we are enabled to correct
them by the following method, which has the advantage of being inde
pendent of the approximate values of the other elements of the orbit.
Vot. IT. F
82 A COMMENTARY ON [SECT. XI.
An exact Determination of the elements of the orbit, 'when we know ap
proximate values of the perihelion distance of the comet, and of the instant
of its passage over the perihelion.
We shall first select three distant observations of the comet; then
taking the perihelion distance of the comet, and the instant of its crossing
the perihelion, determined as above, we shall calculate the three anomalies
of the comet and the corresponding radius-vectors corresponding to the
instants of the three observations. Let v, v', v" be these anomalies, those
which precede the passage over the perihelion being supposed negative.
Also let g, g' g" be the corresponding radius-vectors of the comet ; then
v7 — v, V — v will be the angles comprised by g and g' and by §, g".
Let U be the first of these angles, U' the second. Again, call a, a' a!' the
three observed geocentric longitudes of the comet, referred to a fixed
equinox ; 6, 6', 6" its three geocentric latitudes, the south latitudes being
negative. Let ft, ft', ft'' be the three corresponding heliocentric longi
tudes and *r, w't •&") its three heliocentric latitudes. Lastly call E, E', E"
the three corresponding longitudes of the sun, and R, R', R" its three
distances to the center of the earth.
Conceive that the letter S indicates the center of the sun, T that of the
earth, and C that of the comet, C' that of its projection upon the plane
of the ecliptic. The angle S T C' is the difference of the geocentric lon
gitudes of the sun and of the comet. Adding the logarithm of the cosine
of this angle, to the logarithm of the cosine of the geocentric latitude of
the comet, we shall have the logarithm of the cosine of the angle S T C.
We know, therefore, in the triangle S T C, the side S T or R, the side
S C or g, and the angle S T C, to find the angle C S T. Next we shall
have the heliocentric latitude -a of the comet, by means of the equation
sin. 6 sin. C S T
sin. » =r
sin. C T S
The angle T S C' is the side of a spherical right angled triangle, of
which the hypothenuse is the angle T S C, and of which one of the sides
is the angle ». Whence we shall easily derive the angle T S C7, and con
sequently the heliocentric longitude ft of the comer.
We shall have after the same manner ~', 0'; «•", ft" ; and the values of
ft, ft', ft" will show whether the motion of the comet be direct or retro
grade.
If we imagine (he two arcs of latitude «-, t/, to meet at the pole of the
ecliptic, they would make there an angle equal to ft' — ft; and in the
BOOK I.] NEWTON'S PRINCIPIA. 83
spherical triangle formed by this angle, and by the sides ' — w, - — -a'
it being the semi-circumference, the side opposite to the angle (3f — (3
will be the angle at the sun comprised between the radius-vectors g, and
f'. We shall easily determine this by "spherical Trigonometry, or by the
formula
sin. 2 i V = cos. 2 ^ (w + **') — cos 2 - - (& — 0) cos. • cos. »',
& & £
in which V represents this angle ; so that if we call A the angle of which
the sine squared is
cos 2 - (ft' — ]8) cos. 9 . cos. »',
i)
and which we shall easily find by the tables, we shall have
- ^.' i V = cos. (i . + I ,'+ A) cos. ( \, + i ra'_A).
If in like manner we call V' the angle formed by the two radius-vectors
S) ?"} we have
sin.«iv=cos.(l.+ ^'+
A' being what A becomes, when »', /3' are changed into w", /3".
If, however, the perihelion distance and the instant of the comet's
crossing the perihelion, were exactly determined, and if the observations
were rigorously exact, we should have
V = U, V = U';
But since that is hardly ever the case, we shall suppose
m = U — V ; m' = U' — V.
We shall here observe that the revolution of the triangle S T C, gives
for the angle C S T two different values : for the most part the nature
of the motion of the comets, will show that which we ought to use, and
the more plainly if the two values are very different ; for then the one will
place the comet more distant from the earth, than the other, and it will
be easy to judge, by the apparent motion of the comet at the instant of
observation, which ought to be preferred. But if there remains any un
certainty, we can always remove it, by selecting the value which renders
V and V least different from U and U'.
We next make a second hypothesis in which, retaining the same pas •
snge over the perihelion as before, we shall suppose the perihelion dis
tance to vary by a small quantity ; for instance, by the fiftieth part of
F2
84 A COMMENTARY ON [SECT. XL
its value, and we shall investigate on this hypothesis, the values of U — V,
U' — \T/. Let then
n = U — V ; n' = U' — V.
Lastly, we shall frame a third hypothesis, in which, retaining the same
perihelion distance as in the first, we shall suppose the instant of the pas
sage over the perihelion to vary by a half-day, or a day more or less. In
this new hypothesis we must find the values of
U — VandofU' — V;
which suppose to be
p = U - V, p' = U' — V.
Again, if we suppose u the number by -which we ought to multiply the
supposed variation in the perihelion distance in order to make it the
true one, and t the number by which we ought to multiply the supposed
variation of the instant when the comet passes over the perihelion in
order to make it the true instant, we shall have the two following equa
tions :
(m — n ) u + (m — p ) t = m ;
(m' — n') u + (m' — p') t = m';
whence we derive u and t and consequently the perihelion distance cor
rected, and the true instant of the comet's passing its perihelion.
The preceding corrections suppose the elements determined by the
first approximation, to be sufficiently near the truth for their errors to be
regarded as infinitely small. But if the second approximation should
not even suffice, we can have recourse to a third, by operating upon the ele
ments already corrected as we did upon the first ; provided care be taken to
make them undergo smaller variations. It will also be sufficient to calculate
by these corrected elements the values of U — V, and of U' — V. Call
ing them M, M', we shall substitute them for m, m' in the second mem
bers of the two preceding equations. We shall thus have two new equa
tions which will give the values of u and t, relative to the corrections of
these new elements.
Thus having obtained the true perihelion distance and the true instant
of the comet's passing its perihelion, we obtain the other elements of the
orbit in this manner.
Let j be the longitude of the node which would be ascending if the
motion of the comet were direct, and <p the inclination of the orbit. We
shall have by comparison of the first and last observation,
tan, -a sin. /3' — tan. */ sin. /3 ^
'
tan< -
tan. « cos. jS" — tan. «" cos.
BOOK I.] NEWTON'S PRINCIPIA. 85
tan. -a"
tan. 0 = —. - 7-577 - rr .
sin. (/3"_ j)
Since we can compare thus two and two together, the three observa
tions, it will be more correct to select those which give to the above frac
tions, the greatest numerators and the greatest denominators.
Since tan. j may equally belong to j and <x + j, j being the smallest of
the positive angles containing its value, in order to find that which we
ought to fix upon, we shall observe that <p is positive and less than a right
angle ; and that sin. (/3" — j) ought to have the same sign as tan. -a".
This condition will determine the angle j, and this will be the position
of the ascending node, if the motion of the comet is direct ; but if retro
grade we must add two right angles to the angle j to get the position of
the node.
The hypothenuse of the spherical triangle whose sides are $" — j and
w", is the distance of the comet from its ascending node in the third ob
servation; and the difference between v" and this hypothenuse is the
interval between the node and the perihelion computed along the orbit.
If we wish to give to the theory of a comet all the precision which ob
servations will admit of, we must establish it upon an aggregate of the best
observations ; which may be thus done. Mark with one, two, &c. dashes
or strokes the letters m, n, p relative to the second observation, the third,
&c. all being compared with the first observation. Hence we shaH form
the equations
(m — n ) u + (m — p ) t = m
(m' — n' ) u + (m7 — p' ) t = m'
(m" — n") u + (m" — p") t = m"
&c. = &c.
Again, combining these equations so as to make it easier to determine
u and t, we shall have the corrections of the perihelion distance and of the
instant of the comet's passing its perihelion, founded upon the aggregate
of these observations. We shall have the values of
ft £', 8", &C. », »', w", &C.,
and obtain
. __ tan. * (sin. 3' + sin. B" + &c.) — sin. |8 (tan. «/ + tan. «" + &c.)
"' J ~~ tan. * (cos. B' + cos. B" + &c.) — cos. B (tan. »' + tan. »" + &c.)
_ tan. 'tar' -f- tan. «r" + &c.
*' ~
sin. (/3' — j) + sin. (B" — j) + &c. '
504. There is a case, very rare indeed, in which the orbit of a comet
can be determined rigorously and simply ; it is that where the comet has
been observed in its two nodes. The straight line which joins these
F3
86 A COMMENTARY ON [SECT* XI.
two observed positions, passes through the center of the sun and coincides
with the line of the nodes. The length of this straight line is determined
by the time elapsed between the two observations. Calling T this time
reduced into decimals of a day, and denoting by c the straight line in
question, we shall have (No. 493)
3
1 / T2
= 2 */ ( 9.688724) 2*
Let /3 be the heliocentric longitude of the comet, at the moment of the
first observation ; f its radius- vector ; r its distance from the earth ; and a
its geocentric longitude. Let, moreover, R be the radius of the terrestrial
orbit, at the same instant, and E the corresponding longitude of the sun.
Then we shall have
g sin. (3 = r sin. a — R sin. E ;
g cos. (3 = r cos. a — R cos. E.
Now cr + j3 will be the heliocentric longitude of the comet at the in
stant of the second observation ; and if we distinguish the quantities g, «,
r, R, and E relative to this instant by a dash, we shall have
o' sin. B = R' sin. E' — r' sin. a' ;
g' cos. 3 = R' cos. E' — r' cos. a'.
These four equations give
_ r sin « — R sin. E _ r'sin.a' — R'sin. E'
tan< " ~ rcos.a — Rcos.E r' cos. a' — R' cos. E' '
whence we obtain
, _ R R' sin. (E — E') — R r sin. (« — E')
r sin. (a' — a) — R sin. (a! — E)
We have also
(g 4. £') sin. /3 = r sin. a — r' sin. a' — R sin. E + R' sin. E'
(g -j_ g') cos. j8 = r cos. a — r' cos. a' — R cos. E + R7 cos. E7.
Squaring these two equations, and adding them together, and substitut
ing c for g + g', we shall have
c2 = R2 — 2RR'cos.(E' — E) + R/2
+ 2 r {R' cos. (« — EO — R cos. (a — E)}
+ 2 r' {R cos. (a' — E) — R' cos. (a' — E')l
+ r2— 2rr'cos. (a' — a) + r/2.
If we substitute in this equation for r' its preceding value in terms of r,
we shall have an equation in r of the fourth degree, which can be resolved
by the usual methods. But it will be more simple to find values of r, r'
by trial such as will satisfy the equation. A few trials will suffice for that
purpose.
BOOK I.] NEWTON'S PRINCIPIA. 87
By means of these quantities we shall have /3, g and g'. If v be the
angle which the radius g makes with the perihelion distance called D ;
<r — v will be the angle formed by this same distance, and by the radius g'.
We shall thus have by the equation to the parabola
D D
S =
1s 1
cos. 2 — v sin. 2 — v
;* rw
which give
o
. .
2 r g + s
We shall therefore have the anomaly v of the comet, at the instant of
the first observation, and its perihelion distance D, whence it is easy to
find the position of the perihelion, at the instant of the passage of the
comet over that point. Thus, of the five elements of the orbit of the co
met, four are known, namely, the perihelion distance, the position of the
perihelion, the instant of the comet's passing the perihelion, and the posi
tion of the node. It remains to learn the inclination of the orbit; but for
that purpose it will be necessary to have recourse to a third observation,
which will also serve to select from amongst the real and positive roots of
the equation in r, that which we ought to make use of.
505. The supposition of the parabolic motion of comets is not rigorous ;
it is, at the same time, not at all probable, since compared with the cases
that give the parabolic motion, there is an infinity of those which give the
elliptic or hyperbolic motions. Besides, a comet moving in either a para
bolic or hyperbolic orbit, will only once be visible; thus we may with
reason suppose these bodies, if ever they existed, long since to have dis
appeared ; so that we shall now observe those only which, moving in or
bits returning into themselves, shall, after greater or less incursions into
the regions of space, again approach their center the sun. By the follow
ing method, we shall be able to determine, within a few years, the period
of their revolutions, when we have given a great number of very exact
observations, made before and after the passage over the perihelion.
Let us suppose we have four or a greater number of good observations,
which embrace all the visible part of the orbit, and that we have deter
mined, by the preceding method, the parabola, which nearly satisfies these
observations. Let v, v', v", v'", &c. be the corresponding anomalies;
§1 &'t §"9 f"'> &c- tne radius-vectors. Let also
v' — v = U, v" — v = U', v"' — v = U", &c.
88 A COMMENTARY ON [SECT. XL
Then we shall estimate, by the preceding method with the parabola
already found, the values of U, U', U", &c., V, V, V", &c. Make
m = U — V, m' = U' — V7, m" = U" — V", &c.
Next, let the perihelion distance in this parabola vary by a very small
quantity, and on this hypothesis suppose
n = U — V; n' = U' — V; n" = U" — V", &c.
We will form a third hypothesis, in which the perihelion distance re
maining the same as in the first, we shall make the instant of the comet's
passing its perihelion vary by a very small quantity ; in this case let
p = U — V; p' = U' — V; p" = U" — V"; &c.
Lastly, we shall calculate the angle v and radius g, with the perihelion
distance, and instant over the perihelion on the first hypothesis, supposing
the orbit an ellipse, and the difference 1 — e between its excentricity and
unity a very small quantity, for instance JQ. To get the angle v, in this
hypothesis, it will suffice (489) to add to the anomaly v, calculated in the
parabola of the first hypothesis, a small angle whose sine is
-TJ. (1 — e) tan. — v -| 4 — 3 cos. 2 — v — 6 cos. 4 — v f .
J.U £ \ & £ y
Substituting afterwards in the equation
D
s =
cos'21 '
for v, this anomaly, as calculated in the ellipse, we shall have the corre
sponding radius-vector g. After the same manner, we shall obtain v', g't
v", §", &c. Whence we shall derive the values of U, U', U", &c. and
(by 503) of V, V, V", &c.
In this case let
q = U — V; q' = U' — V ', q" = U" — V", &c.
Finally, call u the number by which we ought to multiply the supposed
variation in the perihelion distance, to make it the true one ; t the number
by which we ought to multiply the supposed variation in the instant over
the perihelion, to make it the true instant ; and s that by which we should
multiply the supposed value of 1 — e, in order to get the true one ; and
we shall obtain these equations :
(m — n) u + (m — p) t + (in — q; s = m;
(m' — n' ) u + (m' — p' ) t + (m' — q') s = m ;
(m" — n") u + (m" — p") t + (m" — q") s = m" ;
(m'" _ n"') u + (m'" — p'") t + (m"' — q'") s = m'";
&c.
BOOK 1.] NEWTON'S PRINCIPIA. 89
We shall determine, by means of these equations, the values of u, t, s;
whence will be derived the true perihelion distance, the true instant over
the perihelion, and the true value of 1 — e. Let D be the perihelion
distance, and a the semi-axis major of the orbit; then we shall have
a = - ; the time of a sidereal revolution of the comet, will be expressed
1 — e
2. / f) 5.
by a number of sidereal years equal to a2 or to (- — -)*> tne mean
\ J^ __ Q/
distance of the sun from the earth being unity. We shall then have
(by 503) the inclination of the orbit and the position of the node.
Whatever accuracy we may attribute to the observations, they will
always leave us in uncertainty as to the periodic times of the comets. To
determine this, the most exact method is that of comparing the observa
tions of a comet in two consecutive revolutions. But this is practicable,
only when the lapse of time shall bring the comet back towards its peri
helion.
Thus much for the motions of the planets and comets as caused by the
action of the principal body of the system. We now come to
506. General methods of determining by successive approximations, the
motions of the heavenly bodies.
In the preceding researches we have merely dwelt upon the elliptic
motion of the heavenly bodies, but in what follows we shall estimate them
as deranged by perturbing forces. The action of these forces requires only
to be added to the differential equations of elliptic motion, whose integrals
in finite terms we have already given, certain small terms. We must deter
mine, however, by successive approximations, the integrals of these same
equations when thus augmented. For this purpose here is a general me
thod, let the number and degree of the equations be what they may.
Suppose that we have between die n_ variables y, y', y", &c. and the
time t whose element d t is constant, the n differential equations
0 =
&c. = &c.
P, Q, P', Q', &c. being functions of t, y, y', &c. and of the differences to
the order i — 1 inclusively, and a being a very small constant coefficient,
which, in the theory of celestial motions, is of the order of the perturb
ing forces. Then let us suppose we have the finite integrals of those
90 A COMMENTARY ON [SECT. XL
equations when Q, Q', &c. are nothing. Differentiating each i — 1
times successively, we shall form with their differentials i n equations by
means of which we shall determine by elimination, the arbitrary constants
c, c', c", &c. in functions of t, y, y', y", &c. and of their differences to the
order i — 1. Designating therefore by V, V, V", &c. these functions
we shall have
c = V; c' = V; c" = V"; &c.
These equations are the i n integrals of the (i — l)th order, which the
equations ought to have, and which, by the elimination of the differences
of the variables, give their finite integrals.
But if we differentiate the preceding integrals of the order i — 1, we
shall have
0 = dV; 0 = d V; 0 = d M" '; &c.
and it is clear that these last equations being differentials of the order i
without arbitrary constants, they can only be the sums of the equations
0 = &c.
each multiplied by proper factors, in order to make these sums exact dif
ferences. Calling, therefore, F d t, F' d t', &c. the factors which ought
respectively to multiply them in order to make 0 = d V ; also in like
manner making H d t, H' d t', &c. the factors which would make On d V,
and so on for the rest, we shall have
&c.
F, F', &c. H, H7, &c. are functions of t, y, y', y", &c. and of their dif
ferences to the order i — 1. It is easy to determine them when V, V7, &c.
are known. For F is evidently the coefficient of -r- ^ in the differential
of V ; F' is the coefficient of — ^ in the same differential, and so on.
Cl L
d ' v d ' y'
In like manner, H, H', &c. are the coefficients of T-f , -,- j > &c. in the
Cl t Cl L
differential of V7. Thus, since we may suppose V, V', &c. known, by dif-
BOOK I.] NEWTON'S PRINCIPIA. 91
(I * ™" * v Cl V
ferentiating with regard to , . _\ , , i _ l , &c. we shall have the
factors by which we ought to multiply the differential equations
0 = |if + P, 0 = ^i' + P', &c.
in order to make them exact differences.
Now resume the differential equations
0 = ^-"-f + P + « . Q ; 0 = -^-yr -f- F + a . Q', &c.
If we multiply the first by F d t, the second by F' d t, and so on, we
shall have by adding the results
0 = d V + a d t {F Q + F' Q' + &c.},
In the same manner, we shall have
0 = d V + a d t JH Q + H' Q' + &c.}
&c.
whence by integration
c _ a/d t {F Q + F Q' + &c.} = V;
c' _ a/d t {H Q + H' Q' + &c.J = V;
&c.
We shall thus have z n differential equations, which will be of the same
form as in the case when Q, Q', &c. are nothing, with this only differ
ence, that the arbitrary constants c, c', c", &c. must be changed into
c_a/dt {FQ+FQ/+&c.}, c — a/dtfHQ+ H'Q'+&c.}&c.
But if. in the supposition of Q, Q', &c. being equal to zero, we eliminate
from the z n integrals of the order i — 1, the differences of the variables
y, y', &c. we shall have n finite integrals of the proposed equations. We
shall therefore have these same integrals when Q, Q', &c. are not zero, by
changing in the first integrals, c, c', &c. into
c — a/d t £FQ + &c.}, c' — «/d t {H Q -f- &c.}&c.
507. If the differentials
d t £F Q + F Q' + &c.J, d t {II Q + H' Q' + &c.J&c.
are exact, we shall have, by the preceding method, finite integrals of the
proposed differentials. But this is not so, except in some particular cases,
of which the most extensive and interesting is that in which they are
linear. Thus let P, P', &c. be linear functions of y, y', &c. and of their
differences up to the order i — 1, without any term independent of these
variables, and let us first consider the case in which Q, Q', &c. are no
thing. The differential equations being linear, their successive integrals
92 A COMMENTARY ON [SECT. XL
are likewise linear, so that c = V, c' = V, &c. being the i n integrals of
the order i — ], of the linear differential equations
V, V, &c. may be supposed linear functions of y y', &c. and of their dif
ferences to the order i — 1. To make this evident, suppose that in the
expressions for y, y', &c. the arbitrary constant c is equal to a determinate
quantity plus an indeterminate d c ; the arbitrary constant c' equal to a
determinate quantity plus an indeterminate 5 c' &c. ; then reducing these
expressions according to the powers and products of d c, d c', &c. we shall
have by the formulas of No. 487
1.2'
2
&c.
Y, Y', f~j — J , &c. being functions oft without arbitrary constants. Sub
stituting those values, in the proposed differential equations, it is evident .
that d c, d c', &c. being indeterminate, the coefficients of the first powers
of such of them ought to be nothing in the several equations. But these
equations being linear, we shall evidently have the terms affected with the
first powers of 8 c, d c', &c. by substituting for y, y', &c. these quantities
respectively
These expressions of y, y', &c. satisfy therefore separately the proposed
equations ; and since they contain the i n arbitraries d c, d c', &c. they are
complete integrals. Thus we perceive, that the arbitraries are under a
linear form in the expressions of y, y', &c. and consequently also in their
differentials. Whence it is easy to conclude that the variables y, y', &c.
and their differences, may be supposed to be linear in the successive inte
grals of the proposed differential equations.
d ' v d * v'
Hence it follows, that F, F', &c. being the coefficients of y— r- , - J. ,
Cl t Cl t
BOOK L] NEWTON'S PRINCIPLE 93
&c. in the differential of V ; H, H', &c. being the coefficients of the same
differences in the differentia] of V, &c. these quantities are functions of
variable t only. Therefore, if we suppose Q, Q', &c. functions of t alone,
the differentials
d t {F Q + F Q' + &c.] ; d t [B Q + IF Q' + &c.$ ; &c.
will be exact.
Hence there results a simple means of obtaining the integrals of any
number whatever n of linear differential equations of the order i, and
which contain any terms a Q, a Q', &c. functions of one variable t, having
known the integrals of the same equations in the case where Q, Q7, &c.
are supposed nothing. For then if we differentiate their n finite integrals
i — 1 times successively, we shall have i n equations which will give, by
elimination, the values of the i n arbitrary constants c, c', &c. in functions
of t, y, y', &c. and of their differences to the i — 1th order. We shall thus
form the i n equations c = V, c' = V, &c. This being done, F, F', &c.
(] i - 1 y (J i - 1 y'
will be the coefficients of -r— j~ , Trntrs &Ct *n ^» ^' ^/J &c* w^
be the coefficients of the same differences in V, and so on. We shall,
therefore, have the finite integrals of the linear differential equations
o = + P + «Q; o = + p' + «Q'; &c.
by changing, in the finite integrals of these equations deprived of their last
terms a Q, a Q', &c. the arbitrary constants c, c', &c. into
c — «/d t £F Q + F' Q'+&c.k c' — a/d t {U Q + H' Q'+&c.| &c.
Let us take, for example, the linear equation
The finite integral of the equation
is (found by multiplying by cos. a t, and then by parts getting
f cos. a t . *—}- = cos. a t ~- + a f sin. a t , j- . d t = cos. a t . -~ +
a sin. a t . y — a 2 f cos. a t . y .'. c = a cos. a t . -*-£; + a sin. a t . y, &c.)
c c'
y = — sin. a t + — cos. a t,
a a
c, c' being arbitrary constants.
94 A COMMENTARY ON [SECT. XI.
This integral gives by differentiation
dy
-r* = c cos. at — c sin. a t.
d t
If we combine this with the integral itself, we shall form two integrals
of the first order
d v
c = a y sin. a t + -r-i cos. a t ;
•
c' = a y cos. at -- r-^- sin. a t ;
and therefore shall have in this case
F = cos. at; H = — sin. a t,
and the complete integral of the proposed equation will therefore be
c c' cc sin. a t _ „ .
y = — sm. a t -4 -- cos. at -- / U d t cos. a t
a a a J
a, cos. a t rf^ , .
-\ -- j Q d t sin. a t.
Hence it is easy to conclude that if Q is composed of terms of the form
K . (m t -4- i) each of these terms will produce in the value of y the
cos. v *
corresponding term
« K sin. .
— 2 ---- ,. (m t + e).
m 2 — a 2 cos. v
If m be equal to a, the term K (m t + t) will produce in y, 1st. the
term — -. — -„ . (a t + «) which being comprised by the two terms
4 a * cos. v
c c' cc 1C t cos
— sin. a t-\ -- cos. at, may be neglected: 2dly. the term + — - — . . '(a t + g)>
a a — 2 a sm.v
+ or — being used according as the term of Q is a sine or cosine. We
thus perceive how the arc t produces itself in the values of y, y', &c. with
out sines and cosines, by successive integrations, although the differentials
do not contain it in that form. It is evident this will take place when
ever the functions F Q, F', Q', &c. H Q, H' Q', &c. shall contain con
stant terms.
508. If the differences
d t [F Q + &c.}, d t { H Q + &c.}
are not exact, the preceding analysis will not give their rigorous integrals.
But it affords a simple process for obtaining them more and more nearly
by approximation when a is very small, and when we have the values of
t
BOOK I.j NEWTON'S PRINCIPIA. 95
y, y7, &c. on the supposition of a being zero. Differentiating these values,
i — 1 times successively, we shall form the differential equations of the
order i — 1, viz.
c = V; c' = V, &c.
d * v d ' v'
The coefficients of j-4- , -.— *- , &c. in the differentials of V, V', &c.
Cl I *J. L
being the values of F, F', &c. H, H', Sec. we shall substitute them in the
differential functions
d t (F Q + F' Q' + &c.) ; d t (H Q + H' Q' + &c) ; &c.
Then, we shall substitute in these functions, for y, y', &c. their first
approximate values, which will make these differences functions of t and of
the arbitrary constants c, c', &c.
Let T d t, T d t, &c. be these functions. If we change in the first
approximate values of y, y', &c. the arbitrary constants c, c', &c. re
spectively into c — a y T d t, c' — a y X d t, &c. we shall have the
second approximate values of those variables.
Again substitute these second values in the differential functions
d t . (F Q + &c.) ; d t (H Q + &c.) &c.
But it is evident that these functions are then what T d t, T' d t, &c.
become when we change the arbitrary constants c, c', &c. into c — af T d t,
c' —ufT d t, &c. Let therefore T/5 T/, &c. denote what T, T, &c.
become by these changes. We shall get the third approximate values of
y, y', &c. by changing in the first c, c', &c. respectively into c — «yX, d t,
c- — yX; d t, &c.
Calling T//} T//, in like manner, what T, X, &c. become when
we change c, c', &c. into c — «y T, d t, c' — «y T/ d t, &c. we shall
have the fourth approximate values of y, y', &c. by changing in the first
approximate values of these variables into c — «y T/7 d t, c' — «y X,/ d t,
&c. and so on.
We shall see presently that the determination of the celestial motions,
depends almost always upon differential equations of the form
0 = ^y + a'y + «Q,
Q being a rational and .integer function of y, of the sine and cosine of
angles increasing proportionally with the time represented by t. The
following is the easiest way of integrating this equation.
First suppose u nothing, and we shall have by the preceding No. a first
value of y.
Next substitute this value in Q, which will thus become a rational and
9G A COMMENTARY ON [SECT. XI.
entire function of sines and cosines of angles proportional to the time.
Then integrating the differential equation, we shall have a second value
of y approximate up to quantities of the order « inclusively.
Again substitute this value in Q, and, integrating the differential equa
tion, we shall have a third approximation of y, and so on.
This way of integrating by approximation the differential equations of
the celestial motions, although the most simple of all, possesses the dis
advantage of giving in the expressions of the variables y, y', &c. the arcs
of a circle (symbols sine and cosine] in the very case where these arcs
do not enter the rigorous values of these variables. We perceive, in
fact, that if these values contain sines or cosines of angles of the order a t,
these sines or cosines ought to present themselves in the form of series, in
the approximate values found by the preceding method ; for these last
values are ordered according to the powers of «. This developement
into series of the sine and cosine of angles of the order a t, ceases to be
exact when, by lapse of time, the arc a t becomes considerable. The ap
proximate values of y, y', &c. cannot extend to the case of an unlimited
interval of time. It being important to obtain values which include both
past and future ages, the reversion of arcs of a circle contained by the
approximate values, into functions which produce them by their develope
ment into series, is a delicate and interesting problem of analysis. Here
follows a general and very simple method of solution.
509. Let us consider the differential equation of the order i,
d v d *~ * v
a being very small, and P and Q algebraic functions of y, -^ , . . . . -, — j— ^ ,
tl L ' tl L
and of sines and cosines of angles increasing proportionally with the time.
Suppose we have the complete integral of this differential, in the case of
a = 0, and that the value of y given by this integral, does not contain the
arc t, without the symbols sine and cosine. Also suppose that in inte
grating this equation by the preceding method of approximation, when a
is not nothing, we have
y = X + t Y + t2 Z + t3 S + &c.
X, Y, Z, &c. being periodic functions of t, which* contain the i arbitraries
c, c', c", &c. and the powers of t in this expression of y, going on to in
finity by the successive approximations. It is evident the coefficients
of these powers will decrease with the greater rapidity, the less is a.
In the theory of the motions of the heavenly bodies, « expresses the order
of perturbing forces, relative to the principal forces which animate them.
BOOK I.] NEWTON'S PRINCIPIA. 97
d ' v
If we substitute the preceding value of y in the function -^H-P-f-aQs
it will take the form k + k' t + k" t2 + &c., k, k', k", &c. being perio
dic functions of t ; but by the supposition, the value of y satisfies the dif
ferential equation
we ought therefore to have identically
0 = k + k' t + k" t 2 + &c.
If k, k', k", &c. be not zero this equation will give by the inversion of
series, the arc t in functions of sines and cosines of angles proportional to
the time t. Supposing therefore a to be infinitely small, we shall have t
equal to a finite function of sines and cosines of similar angles, which is
impossible. Hence the functions k, k', &c. are identically nothing.
Again, if the arc t is only raised to the first power under the symbols
sine and cosine, since that takes place in the theory of celestial motions,
the arc will not be produced by the successive differences of y. Substi
tuting, therefore, the preceding value of y, in the function ~ ^+P+« . Q,
the function of k + k' t + &c. to which it transforms, will not contain
the arc t out of the symbols sine and cosine, inasmuch as it is already con
tained in y. Thus changing in the expression of y, the arc t, without the
periodic symbols, into t — 0, 6 being any constant whatever, the function
k + k' t + &c. will become k + k' (t — 6) + &c. and since this last
function is identically nothing by reason of the identical equations k =r 0
k' = 0, it results that the expression
y = X + (t — 6} Y + (t — 6) 2 Z + &c.
also satisfies the differential equation
0 = ai? + p + « Q-
Although this second value of y seems to contain i + 1 arbitrary con
stants, namely, the i arbitrages c, c', c", &c. and 6t yet it can only have i
distinct ones. It is therefore necessary that by a proper change in the
constants c, c', &c. the arbitrary 6 be made to disappear, and thus the
second value of y will coincide with the first. This consideration will fur
nish us with the means of making disappear the arc of a circle out of the
periodic symbols.
Give the following form to the second expression for y :
y - X + (t - 0 . R.
V0l. II. Cr
98 A COMMENTARY ON [SECT. XL
Then supposing 6 to disappear from y, we have
and consequently
('-'> (irr)-
Differentiating successively this equation we shall have
whence it is easy to obtain, by eliminating R and its differentials, from the
preceding expression of y,
(t-4)' /d'X
"
-Xt i , ,
'} l~dr; + TT"" • I dT^) H
X is a function of t, and of the constants, c, c', c", &c. and since these
constants are functions of 6, X is a function of t and of 6, which we can
represent by <f> (t, 6). The expression of y is by Taylor's Theorem
the developement of the function <p (t, 6 + t — <5), according to the powers
of t — 6. We have therefore y = <p (t, t). Whence we shall have y by
changing in X, 6 into t. The problem thus reduces itself to determine
X in a function of t and 6, and consequently to determine c, c', c", &c.
in functions of 6.
To solve this problem, let us resume the equation
y = X + (t — 6) . Y + (t — /) 2 . Z + &c.
Since the constant 6 is supposed to disappear from this expression of y,
we shall have the identical equation
. . .(a)
Applying to this equation the reasoning which we employed upon
0 = k + k' t + k" t2 + &c.
we perceive that the coefficients of the successive powers of t — 6 ought
to be each zero. The functions X, Y, Z, &c. do not contain 6, inasmuch
as it is contained in c, c', &c. so that to form the partial differences
(i?) , ( 1X> , (*?5 > &c. it is sufficient to make c, c', &c. vary in
V d 6 ) \ d 0 / Yd* *• / -
these functions, which gives
X
d d
_ (.c ( xc ,}
- \d c )d 6 H Vd c') d 4 + \d c") d d
BOOK I.] NEWTON'S PRINCIPIA. 99
a YX /a YX dc /aY\c /a
\c , / N
Vdo + \dc")~dl
&C. zr &C.
Again, it may happen that some of the arbitrary constants c, c', c", &c.
multiply the arc t in the periodic functions X, Y, Z, &c. The differentia
tion of these functions relatively to 6, or, which is the same thing, relatively
to these arbitrary constants, will develope this arc, and bring it from without
the symbols of the periodic functions. The differences ( , --), (
\tl D / \
V &c. will be then of this form :
&C.
X', X", Y', Y", Z', Z", &c. being periodic functions of t, and containing
moreover the arbitrary constants c, c', c", &c. and their first differences
divided by d 6, differences which enter into these functions only under a
linear form ; we shall have therefore
= Y/ + ' Y// + (fc ~ v Y"
TIT = z' + * z" + (t — o z-
ate,
Substituting these values in the equation (a) we shall have
0 = X' + 6 X;/ — Y
+ (t — 6) iY' + 6 Y" + X" — 2 Z}
+ (t — 6) MZ' + * Z" + Y" — 3 S} + Sec. ;
whence we derive, in equalling separately to zero, the coefficients of the
powers of t — 6,
0 = X' + 0 X" — Y
0 = Y' + & Y" + X" — 2 Z
0 = Z' + 0Z" + Y" — 3 Sj
&c.
.G 2
100 A COMMENTARY ON [SECT. XL
If we differentiate the first of these equations, i — 1 times successively
relatively to t, we shall thence derive as many equations between the
quantities c, c', c", &c. and their first differences divided by d 6. Then
integrating these new equations relatively to 6, we shall obtain the con
stants in terms of 6.
Inspection alone of the first of the above equations will almost always
suffice to get the differential equations in c, c', c", &c. by comparing se
parately the coefficients of the sines and cosines which it contains. For
it is evident that the values of c, c', &c. being independent of t, the dif
ferential equations which determine them, ought, in like manner, to be in
dependent of it. The simplicity which this consideration gives to the pro
cess, is one of its principal advantages. For the most part these equations
will not be integrable except by successive approximations, which will
introduce the arc 6 out of the periodic symbols, in the values of c, c', &c.
at the same time that this arc does not enter the rigorous integrals. But
we can make it disappear by the following method.
It may happen that the first of the preceding equations, and its i — 1
differentials in t, do not give a number i of distinct equations between the
quantities c, c', c", &c. and their differences. In this case we must have
recourse to the second and following equations.
When we shall have thus determined c, c', c", &c. in functions of d,
we shall substitute them in X, and changing afterwards 6 into t, we shall
obtain the value of y, without arcs of acircle^or free from periodic symbols,
when that is possible.
510. Let us now consider any number n of differential equations.
o = ^-*r + P + « Q' ;
&c.
P, Q, P', Q' being functions of y, y', &c. of their differentials to the order
i — 1, and of the sines and cosines of angles increasing proportionally
with the variable t, whose difference is constant. Suppose the approximate
integrals of these equations to be
y - X + t Y + t2 Z + t3 S + &c.
y' = X, + t Y, + t2 Z, + t 3 S, + &c.
X, Y, Z, &c. X,, Y;, Z,, &c. being periodic functions of t and containing
i n arbitrary constants c, c', c", &c. We shall have as in the preceding
No.
BOOK I.] NEWTON'S PRINCIfiA; 101
0 = X' + dX" — Y;
0 = Y' + 6 Y" + X" — 2 Z;
0 = Z + 6 Z" + Y" — 3 S ;
&c.
The value of y' will give, in like manner, equations of this form
0 = X/ + *X," — Y,;
0 = Y/ + 0Y," + X/' — g Z/;
&c.
The values of y", y'", &c. will furnish similar equations. We shall
determine by these different equations, selecting the most simple and
approximable, the values of c, c', c", &c. in functions of 6. Substituting
these values in X, X', &c. and then changing 6 into t, we shall have the
values of y, y', &c. independent of arcs free from periodic symbols when
that is possible.
511. Let us resume the method already exposed in No. 506. It theucc
results that, if instead of supposing the parameters c, c', c", &c. constant,
we make them vary so that we have
d c = — a d t [F Q + F' Q' + &c} ;
d c' = — « d t SH Q + H' Q' + &c.J ;
we shall always have the i n integrals of the order i — 1,
c = V; c? = V; c" = V" ; &c.
as in the case of a = 0. Whence it follows that not only the finite in
tegrals, but also all the equations in which these enter the differences
inferior to the order i, will preserve the same form, in the case of
a = 0, and in that where it is any quantity whatever; for these equations
may result from the comparison alone of the preceding integrals of the
order i — 1. We can, therefore, in the two cases equally differentiate
i — 1 times successively the finite integrals, without causing c, c', &c. to
vary ; and since we are at liberty to make all vary together, there will
thence result the equations of condition between the parameters c, c', &c.
and their differences.
In the two cases where a = 0, and a = any quantity whatever, the
values of y, y', &c. and of their differences to the order i — 1 inclusively,
are the same functions of t and of the parameters c, c', &c. Let Y be any
function of the variables y, y', y", &c. and of their differentials inferior to
the order i — 1, and call T the function of t, which it becomes, when we
substitute for these variables and their differences their values in t. We
can differentiate the equation Y = T, regarding the parameters c, c', &c.
constant ; we can only, however, take the partial difference of Y relatively
G3
102
• A COMMENTARY ON [SECT. XL
to one only or to many of the variables y, y', &c. provided we suppose
what varies with these," to vary also in T. In all these differentiations, the
parameters c, c', c", «:c. may always be treated as constants ; since by
substituting for y, y', &c. and their differences, their values in t, we shall
have equations identically zero in the two cases of" nothing and of a any
quantity whatever.
When the differential equations are of the order i — 1, it is no longer
allowed, in differentiating them, to treat the parameters c, c', &c. as con
stants To differentiate these equations, consider the equation <p = 0, 9
bein- a differential function of the order i - 1, and which contains the
parameters c, c', c", &c. Let d f be the difference of this function taken
in regarding c, c', &c. constant, as also the differences d > • 1 y, d ' - ' y', &c.
Let S be the coefficient of £& in the entire difference of f. Let S'
be the coefficient of £^ in this same difference, and so on. The e, ua-
tion 9 = 0 when differentiated will give
Substituting for £& its value - d t IP + . QJ ; for ££ i* value
_ d t {P' + « Q'S &c. we shall have
_ d t JS P + S' F + &c.} — « d t [S Q + S' Q' + &c.} . (t)
In the supposition of « = 0, the parameters c, c', c", &c. are constant.
We have thus
0 = a ? — d t £S P + S' F + &c.}
If we substitute in this equation for c, c', c", &c. their values V, V, V-,
&c. we shall have differential equations of the order i - 1 , without arbi-
traries, which is impossible, at least if this equation is to be id,:
nothing. The function
3 p — d t {S P + S' F + &c.J
becoming therefore identically nothing by reason of equations c : : V,
c - V &c. and since these equations hold still, when the parameters
c, c", C",' &c. are variable, it is evident, that in this case, the preceding
BOOK L] NEWTON'S PRINCIPIA. 103
function is still identically nothing. The equation (t) therefore will be
come
— a d t {S Q + S' Q' + &c.} ....... (X)
Thus we perceive that to differentiate the equation <p = 0, it suffices to
vary the parameters c, c', &c. in <p and the differences d1-1 y, d i~1 y',
&c. and to substitute after the differentiations, for — a Q, a Q', &c. the
d ' v d ' v'
quantities^, -4. , &c.
Let 4 = 0, be a finite equation between y, y', Sec. and the variable t. If
we designate by d 4, d z 4, &c. the successive differences of 4, taken in
regarding c, c', &c. as constant, we shall have, by what precedes, in that
case where c, c', &c. are variable, these equations :
4 = 0; 54 = 0; a2 4 = 0 ...... a1-1 4 = o;
changing therefore successively in the equation (x) the function <p into 4,
d 4> ^ 2 4> &c. we shall have
°=(T±\
\u c /
d
Thus the equations 4 = 0, 4' = 0, &c. being supposed to be the n
finite integrals of the differential equations
d ! v
d t1
&c.
we shall have i n equations, by means of which we shall be able to de
termine the parameters c, c', c", &c. without which it would be necessary
for that purpose to form the equations c = V, c' = V, &c. But when
the integrals are under this last form, the determination will be more
simple.
512. This method of making the parameters vary, is one of great utility
G3
A COMMENTARY ON [SECT. XI.
in analysis and in its applications. To exhibit a new use of it, let us take
the differential equation
d ' v
0 = TP + p'
P being a function of t, y, of their differences to the order i — ], and of
the quantities q, q', &c. which are functions of t. Suppose we have the
finite integral of this differential equation of the supposition of q, q', &c.
being constant, and represent by p = 0, this integral, which shall contain
i arbitraries c, c', &c. Designate by d <p, 3 2 p, 8 3 p, &c. the successive differ
ences of p taken in regarding q, q', &c. constant, as also the parameters
c, c', c", &c. If we suppose all these quantities to vary, the differences of
p will be
making therefore
0 = (d-D d « + CH) o " + «*• + d q + -J ^ +
a p will be still the first difference of <p in the case of c, c', &c. q, q', &c.
being variable. If we make, in like manner,
•9 z 9) & 3 P) ..... ^ 5 p will likewise be the second, third, &c. differences of
<p when c, c', &c. q, q', &c. are supposed variable.
Again in the case of c, c', &c. q, q', &c. being constant, the differential
equation
d ' v
° = Si? + p'
is the result of the elimination of the parameters c, c', &c. by means of
the equations p = 0, 8 <p = 0, 8 2 p = 0, . . . . d ! p = 0. Thus, these
last equations still holding good when q, q', &c. are supposed variable, the
equation <p = 0 will also satisfy, in this case, the proposed differential
equation, provided the parameters c, c', &c. are determined by means
of the i preceding differential equations ; and since their integration
gives i arbitrary constants, the function <p will contain these arbitraries,
and the equation <p = 0 will be the complete integral of the proposed
equation.
BOOK I.] NEWTON'S PRINCIPIA. 105
This method, the variation of parameters, may be employed with ad
vantage when the quantities q, q', &c. vary very slowly. Because this
consideration renders the integration by approximation of the differential
equations which determine the variables c, c', c", &c. in general much
easier.
513. Second Approximation of Celestial Motions.
Let us apply the preceding method to the perturbations of celestial
motions, in order thence to obtain the most simple expressions of their
periodical and secular inequalities. For that purpose let us resume the
differential equations (1), (2), (3) of No. 471, which determine the relative
motion of p about M. If we make
R = l '
+ y'2 + z'2)^ (X"2 + y"2 + z"2)*
-4- <tc -
r oii^"
t*
X being by the No. cited equal to
» (*'
f (x" _ x') 2 + (y" — yT + (z" — z') 2} B
If, moreover, we suppose M + ^ — m and
i" — x)2+(y" —
r + &c.
s = V x2 + y2 + z1
S' = V x'2 + y/2 + z
we shall have
d 2 z inz
" dt2 »
(P)
_
The sum of these three equations multiplied respectively by d x, d y, d z
gives by integration
2m m
+ ~
~ -- ~ (Q)
the differential d R being only relative to the coordinates x, y, z of the
body ft, and a being an arbitrary constant, which, when R = 0, becomes
by No. 499, the semi-axis major of the ellipse described by ft about
M.
106 A COMMENTARY ON [SECT. XI.
The equations (P) multiplied respectively by x, y, z and added to the
integral (Q) will give
We may conceive, however, the perturbing masses /, ///', £c. multi
plied by a coefficient «, and then the value of g will be a function of the
time t and of «. If we develope this function according to the powers of a,
and afterwards make a = 1, it will be ordered according to the powers
and products of the perturbing masses. Designate by the characteristic
8 when placed before a quantity, this differential of it taken relatively to «,
and divided by d «. When we shall have determined <§ g in a series or
dered according to the powers of a, we shall have the radius g by multi
plying this series by d «, then integrating it relatively to «, and adding to
the integral a function of t independent of «, a function which is evidently
the value of g in the case where the perturbing forces are nothing, and
where the body p describes a conic section. The determination of g re
duces itself, therefore, to forming and integrating the differential equation
which determines d g.
For that purpose, resume the differential equation (R) and make for the
greater simplicity
d Rx /d R
differentiating this relatively to «, we shall have
Call d v the indefinitely small arc intercepted between the two radius-
vectors g and g + d g ; the element of the curve described by //. around M
will be V dg 2 + g*d\\ We shall' thus have
clx2 + dy2 + dz2 — d §z + g2d v2,
and the equation (Q) will become
dt2 g a
Eliminating — from this equation by means of equation (R) we shall
ft
have
^Tt1" : Tt^" + T + s R
whence we derive, by differentiating relatively to a,
d t2 d t2 r* *~ s a ~~ s'
BOOK I.] NEWTON'S PRINCIPIA 107
If we substitute in this equation for — ^-^ its value derived from equa
tion (S), we shall have
By means of the equations (S), (T), we can get as exactly as we wish the
values of B g and of d v. But we must observe that d v being the angle
intercepted between the radii g and g + d & the integral v of these angles
is not wholly in one plane. To obtain the value of the angle described
round M, by the projection of the radius-vector g upon a fixed plane, de
note by v, , this last angle, and name s the tangent of the latitude of ^ above
this plane ; then g (I + s 2) ~ ¥ will be the expression of the projected ra
dius-vector, and the square of the element of the curve described by p,
will be
r+V2 + df2 + (iT^r5
But the square of this element is also g2 d v2 + d g2; therefore we have,
by equating these two expressions
.,
We shall thus determine d vy by means of d v, when s is known.
If we take for the fixed plane, that of the orbit of p at a given epoch,
s an{i °J wi]l evidently be of the order of perturbing forces. Neglecting
d v
therefore the squares and the products of these forces, we shall have
v = v, . In the Theory of the planets and of the comets, we may neglect
these squares and products with the exception of some terms of that
order, which particular circumstances render of sensible magnitude, and
which it will be easy to determine by means of the equations (S) and (T).
These last equations take a very simple form, when we take into account
the first power only of the disturbing forces. In fact, we may then con
sider 8 i and d v as the parts of g and v due to these forces ; d II, d. g R'
are what R and g R' become, when we substitute for the coordinates of
the bodies their values relative to the elliptic motion : We may designate
them by these last quantities when subjected to that condition. The
equation (S) thus becomes,
0 = + =-«- + 2/rf R + , R'.
108 A COMMENTARY ON [SECT. XL
The fixed plane of x, y being supposed that of the orbit of ^ at a given
epoch, z will be of the order of perturbing forces : and since we may
neglect the square of these forces, we can also neglect the quantity
Z \dz)' Moreover, the radius g differs only from its projection by quan
tities of the order z 2. The angle which this radius makes with the axis
of x, differs only from its projection by quantities of the same order.
This angle may therefore be supposed equal to v and to quantities nearly
of the same order
x = § cos. v ; y = g sin. v ;
whence we get
d R
and consequently g . R' = s ^fr • li is easy to perceive by differentia
tion, that if we neglect the square of the perturbing force, the preceding
differential equation will become, by means of the two first equations (P)
'/y^'/'R + Kffil-y/'d'fr/'H +,(ffi }
/x d y — y d xx
v ai — )
In the second member of this equation the coordinates may belono- to
elliptic motion ; this gives ^7? ( — constant and equal to V~m a(l — e2),
a e being the excentricity of the orbit of p. If we substitute in the ex
pression of § 8 § for x and y, their values g cos. v and § sin. v, and for
x d y _ v d x _
cj t - , the quantity v" ,«- a (1 — e2) ; finally, if we observe that
by No. (480)
m = n 2 a
we shall have
(" a cos. v/n d t . s sin. v { 2fd R + s (-.—} \
Vde'-»
} )
e'-» V
j
(X)
-«sin.v/ndt.ecos.v2/rfR + f ()
o o ^: -- - - =^ - 5 -
m V 1 — e2
The equation (T) gives by integration and neglecting the square of
perturbing forces,
2 g d . a g + d s . ds 3 a rr , „ , 2 a , /d Rx
— - - i — PT-= — - + - - // n d t . d R H -- fn d t. g ( —, — )
a * n d t ^ m J- m J s \ d /
.,,.
BOOK I.] NEWTON'S PRINCIPIA. 109
This expression, when the perturbations of the radius-vector are known,
will easily give those of the motion of p in longitude.
It remains for us to determine the perturbations of the motion in lati
tude. For that purpose let us resume the third of the equations (P):
integrating this in the same manner as we have integrated the equation
(S), and making z = f 8 s, we shall have
r -, . . /d R\ . r j ^ Vd R\
a cos. vyn d t.^sin. v [—. — ) — asm. vyndt.gcos. vf-r— )
a s = - dz/ — - Uz ; (Z)
m v 1— e2
6 s is the latitude of /a above the plane of its primitive orbit: if we wish
to refer the motion of /A to a plane somewhat inclined to this orbit, by
calling s its latitude, when it is supposed not to quit the plane of the
orbit, s + 5 s will be very nearly the latitude of & above the proposed
plane.
514. The formulas (X), (Y), (Z) have the advantage of presenting the
perturbations under a finite form. This is very useful in the Cometary
Theory, in which these perturbations can only be determined by quad
ratures. But the excentricity and inclination of the respective orbits of
the planets being small, permits a developement of their perturbations
into converging series of the sines and cosines of angles increasing pro
portionally to the time, and thence to make tables of them to serve for
any times whatever. Then, instead of the preceding expressions of 8 g,
8 s, it is more commodious to make use of differential equations which
determine these variables. Ordering these equations according to the
powers and products of the excentricities and inclinations of the orbits,
we may always reduce the determination of the values of B g, and of 8 s
to the integration of equations of the form
equations whose integrals we have already given in No. 509. But we
can immediately reduce the preceding differential equations to this simple
form, by the following method.
Let us resume the equation (R) of the preceding No., and abridge it
by making
It thus becomes
'
110 A COMMENTARY ON [SECT. XI.
In the case of elliptic motion, where Q = 0, g 2 is by No. (488) a func
tion of e cos. (n t + t — •*•), a e being the excentricity of the orbit, and
n t + e — -a the mean anomaly of the planet p. Let e cos. (n t + £ — w)
= u, and suppose £ 2 = <p (u) ; we shall have
In the case of disturbed motion, we can still suppose p, 2 = <f> (u), but
u will no longer be equal to e cos. (n t + t — *r). It will be given by
the preceding differential equation augmented by a term depending upon
the perturbing forces. To determine this term, we shall observe that if
we make u = 4/ (g 2) we shall have
•4/ (e2) being the differential of -fy (e.2) divided by d.*2 and ^" (g2) the
d2 ?*
differential of -4/ (f2) divided by d.f2. The equation (R') gives -jf^-
equal to a function of g plus a function depending upon the perturbing
force. If we multiply this equation by 2 f d f, and then integrate it, we
2 1 2
shall have ^U — f- equal to a function of g plus a function depending upon
d2. e2 e2 d e2 .
the perturbing force. Substituting these values of ~2 and of , 8 - in
the preceding expression of -. — - + n 2 u, the function of & which is in
dependent of the perturbing force will disappear of itself, because it is
identically nothing when that force is nothing. We shall therefore have
d2u d2. e2 p2de2
the value of - — - + n2 u by substituting for • ' , and , 2 , the parts
Ci 1 Q C
of their expressions which depend upon the perturbing force. But re
garding these parts only, the equation (R') and its integral give
d2-?2- 20-
"" '
Wherefore
. § d s-
Again, from the equation u = <p (§ 2), we derive d u — 2 g d g ty (f 2) ;
this f * = <p (u) gives 2 p, d f = d u. ff (u) and consequently
BOOK 1.1 NEWTON'S PRINCIPIA. Ill
4 (f2) = p~fifi*
Differentiating this last equation and substituting <ff (u) for - j ^ , we
shall have
p" (u) being equal to ' ' * ^ • , in the same way as <f>' (u) is equal to
. ' P 'u/). . This being done ; if we make
d u
u = e cos. (n t + £ — <ar) 4* ^ u»
the differential equation in M will become
and if we neglect the square of the perturbing force, u may be supposed
equal to e cos. (n t + 2 — «•), in the terms depending upon Q.
The value of - found in No. (485) gives, including quantities of the
a
order e 3
, = .{l + e'-n(l-f e»)-n'-f u'}
whence we derive
^ =aa|l + 2e«— 2u(l— i e2) — u2— u3 j = p (u).
If we substitute this value of p (u) in the differential equation in d u,
and restore to Q its value 2 / d R + g (-jr) » and e cos- (n l + s — w)
for u, we shall have including quantities of the order e 3,
— i-fl + 4 e2— ecos. (nt + £— ») — --e« cos. (2 n t + 2 a —
az 4 4t
When we shall have determined 5 u by means of this differential equa-
112 A COMMENTARY ON [SECT. XI.
tion, we shall have 3 g by differentiating the expression of g, relative to
the characteristic <3, which gives
f 3 9 1
dg = — adu< 1 + -e2+2ecos. (n t + £— »)+ - e2 cos.(2nt+ 2s— 2tr) V.
This value of 5 g will give that of d v by means of formula (Y) of the
preceding number.
It remains for us to determine d s ; but if we compare the formulas (X)
and (Z) of the preceding No. we perceive that d g changes itself into 8 s
by substituting (^-) for 2fdR + g frp-1 in its expression. Whence
it follows that to get d s, it suffices to make this change in the differential
equation in (5 u, and then to substitute the value of 5 u given by this equa
tion, and which we shall designate by d u', in the expression of 8 g. Thus
we get
o =r£jr + ****'
"a2!1 +4et~~ e cos. (n t+ e— w) — ;j- e*cos.(2nt-f 2«— 2
3s= — aSu' 1 1 + ~e* +2 e cos. (nt + s— w)+ ^e2cos.(2nt-f 2e— 2») j
The system of equations (X7), (Y), (Z') will give, in a very simple
manner, the perturbed motion of IL in taking into account only the first
power of the perturbing force. The consideration of terms due to this
power being in the Theory of Planets very nearly sufficient to determine
their motions, we proceed to derive from them formulas for that purpose.
515. It is first necessary to develope the function R into a series. If
we disregard all other actions than that of ,«- upon ^', we shall have by (513}
R _. .^(xx'+yy'+zzO __ ^ ___
(x/2 + y/2 + z'2)^ f(x' — x)2+ (y'_ y)2+ (z' — z)2^ '
This function is wholly independent of the position of the plane of x,
y ; for the radical V (x' — x) 2 + (y' — y)2+ (z' — z) 2, expressing the
distance of n, ,«/, is independent of the position ; the function x 2 + y 2
-f z 2 + x' z + y' 2 + T! 2 — 2 x x' — 2 y y' — 2 z z' is in like manner in
dependent of it. But the squares x2 + y2 + z2 and x/2 + y/2 + z'2
of the radius- vectors, do not depend upon the position ; and therefore the
quantity x x' + y y' + z z' does not depend upon it, and consequently
BOOK I.] NEWTON'S PRINCIPIA. 113
R is independent of the position of the plane of x, y. Suppose in this
function
we shall then have
K - ^gg/cos-
At — ~
x = f cos. v ; y = f sin. v ;
x' = g'cos. v'; y' — P' sin. \f ;
^
(/ 2 + z' 2) 2 g z—2 £ f cos. ( v' — v) + g' 2 + (z'— z) 2] *
The orbits of the planets being almost circular and but little inclined
to one another, we may select the plane of x, y, so that z and z' may be
very small. In this case g and gf are very little different from the semi-
axis-majors a, a' of the elliptic orbits, we will therefore suppose
g = a(l + u,); f = a'(l + u/);
u, and u/ being small quantities. The angles v, v' differing but little
from the mean longitudes n t' + «, n' t + t', we shall suppose
v = n t + s + v,; v' = n' t + «' + v/;
v' and v/ being inconsiderable. Thus, reducing R into a series ordered
according to the powers and products of u,, v,, z, u/, v/, and z', this series
will be very convergent. Let
~9 cos. (n' t — n t + J — 0 — {a 2 — 2 a a' cos. (n' t — n t + i'— e)-f a'2} ~
= g A w + A <« cos. (n7 1 -— n t + •' -— 0 -r- A ® cos. 2 (n' t — n t +/— *)
+ A W cos. 3 (n7 t — n t + e' — «) + &c. ;
We may give to this series the form £ 2 A W cos. i (n' t — n t + i' _ *),
the characteristic 2 of finite integrals, being relative to the number i, and
extending itself to all whole numbers from i = — co to i = oo ; the value
i = 0, being comprised in this infinite number of values. But then we
must observe that A <-« = A (i). This form has the advantage of serving
to express after a very simple manner, not only the preceding series, but
also the product of this series, by the sine or the cosine of any angle
ft + •&; for it is perceptible that this product is equal to
This property will furnish us with very commodious expressions fcr
the perturbations of the planets. Let in like manner
[a * — 2 a a' cos. (n t — n t -f- 1 — ») -f a' 2] ~" *
= ^ 2 B ' cos. i (n t — n t + « — t) ;
B<-'> being equal to B <". This being done, we shall have by (483)
VOL. II.
114 A COMMENTARY ON [SECT. XI.
ttf
R = —- . 2 A W cos. i (n' t — n t + «' — e)
0
+ £u, 2 a(ddAa-)cos. i (n' t _ n t + •' - «)
s- i (n' t — n t + i' — 0
u
— -- (v/ — v,) 2 . i A « sin. i (n' t — n t + %' --
- . u,«. 2-a2-™s- i (n' t - n t + •' - 0
S.i(n't — nt + ,' — ,)
s. i n' t - n t + *' - ,)
(v/ - v') u/ 2' J a/ - Sin< i (n' t - n t + .' - 0
— ~ (v/ — v,) 2 . 2 . i * A (l> cos. i (n' t — n t + s' — 0
'i1
/// z z' 3 /i' a T! z
+ —,s --- f^-4- cos. (n' t — n t + •' — •)
„/ fy' _ y\Z
+ ' ^ ' 2 B W cos. i (n' t — n t + s' — 0
+ &c.
If we substitute in this expression of R, instead of u/5 u/, v/} v/, z and z7,
their values relative to elliptic motion, values which are functions of sines
and cosines of the angles n t + s, n' t + tr and of their multiples, R will
be expressed by an infinite series of cosines of the form <«•' k cos. (i n' t
— i n t + A), i and i' being whole numbers.
It is evident that the action of (J>", (*'", &c. upon p will produce in R
terms analogous to those which result from the action of /«/, and we shall
obtain them by changing in the preceding expression of R, all that relates
to /*', in the same quantities relative to &"> i"/", &c.
Let us" consider any term (i! k cos. (i' n' t — i n t + A) of the expres
sion of R. If the orbits were circular, and in one plane we should
have i' = i. Therefore i' cannot surpass i or be exceeded by it, except
by means of the sines or cosines of the expression for u/} v/9 z, u/, v/, z'
which combined with the sines and cosines of the angle n't — nt + l/ — f
BOOK I.] NEWTON'S PRINCIPIA. 115
and of its multiples, produce the sines and cosines of angles in which i'
is different from i.
If we regard the excentricities and inclinations of the orbits as veiy
small quantities of the first order, it will result from the theorems of
(48 1 ) that in the expressions of u/5 v,, z or g s, s being the tangent of the
latitude of p, the coefficient of the sine or of the cosine of an angle such
as f. (n t + 2), is expressed by a series whose first term is of the order f ;
second term of the order f + 2 ; third term of the order f + 4 and so
on. The same takes place with regard to the coefficient of the sine or of
the cosine of the angle f ' (n' t + /) in the expressions of u/, v/, z'. Hence
it follows that i, and i' being supposed positive and i' greater than i, the
coefficient k in the term m' k cos. (i' n' t — i n t + A) is of the order
i' — i, and that in the series which expresses it, the first term is of the
order V — i the second of the order V — i -f 2 and so on ; so that the
series is very convergent. If i be greater than i', the terms of the series
will be successively of the orders i — i', i — V -j- 2, &c.
Call -a the longitude of the perihelion of the orbit of p and 6 that of its
node, in like manner call •**' the longitude of the perihelion of ,«/, and ¥
that of its node, these longitudes being reckoned upon a plane inclined
to that of the orbits. It results from the Theorems of (481), that in the
expressions of u/5 v,, and z, the angle n t + s is always accompanied by
— » or by — 6 1 and that in the expressions of u/, v/, and z', the angle
n' t + t' is always accompanied by — «•', or by — 6' ; whence it follows
that the term (t! k cos. (i' n' t — i n t + A) is of the form
y! k cos. (i n7 1 — i n t -f- i' e — is — g - — g7 J — g" 6 — g"' </),
g, g', g", g"' being whole positive or negative numbers, and such that
we have
0 = i' - i — g — g' — g" — g"'.
It results also from this that the value of R, and its different terms are
independent of the position of the straight line from which the longitudes
are measured. Moreover in the Theorems of (No. 481) the coefficient of
the sine and cosine of the angle •», has always for a factor the excentricity e
of the orbit of p ; the coefficient of the sine and of the cosine of the angle
2 9) has for a factor the square e2 of this excentricity, and so on. In like
manner, the coefficient of the sine and cosine of the angle d, has for its
factor tan. \ tp, <p being the inclination of the orbit of /A upon the fixed
plane. The coefficient of the sine, and of the cosine of the angle 2 6, has for
its factor tan.2 \ <p, and so on. Whence it results that the coefficient k has for
its factor, e *. e' s'. tan. g" (£ <p ) tan. g/// (± <f>'} ; the numbers g, g7, g", g"' being
H2
116 A COMMENTARY ON [SECT. XI.
taken positively in the exponents of this factor. If all these numbers are
positive, this factor will be of the order i7 — i, by virtue of the equation
0 = i'__i_g_g'_g"_ g"';
but if one of them such as g, is negative and equal to — g, this factor
will be of the order i' — i + 2 g. Preserving, therefore, amongst the
terms of R, only those which depending upon the angle i' n7 t — i n t are of
the order i' — i, and rejecting all those which depending upon the same
angle, are of the order i' — i + 2, i' — i + 4, &c. ; the expression of
R will be composed of terms of the form
H e 8. e7 £' tan. *" ( ~ p) tan. *'". ( i ?/) cos. (i' n' t — i n t + i' *'
_ i f _ g. „ — g'. .' — g". 8 - g777. 0'),
H being a coefficient independent of the excentricities, and inclinations
of the orbits, and the numbers g, g', g", g'" being all positive, and such
that their sum is equal to i' — i.
If we substitute in R, a (1 + uy), instead of s, we shall have
d Rx /d R
If in this same function, we substitute instead of u7, v7 and z, their values
given by the theorems of (481), we shall have
/d RN _ /d R-N .
Vd v/~J>d J'
provided that we suppose s — », and s — ^ constant in the differential of
R, taken relatively to z ; for then u/} v/ and z are constant in this differ
ential, and since we have v = n t -f s + v/} it is evident that the preced
ing equation still holds. We shall, therefore, easily obtain the values
and of f^r^V which enter into the differential equations of
the preceding numbers, when we shall have the value of R developed
into a series of angles increasing proportionally to the time t. The dif
ferential d R it will be in like manner easy to determine, observing to vary
in R the angle n t, and to suppose n7 t constant ; for d R is the difference
of R, taken in supposing constant, the coordinates of //, which are func
tions of n7 t.
516. The difficulty of the developement of R into a series, may be
reduced to that of forming the quantities #«, B *\ and their differences
taken relatively to a and to a7. For that purpose consider generally the
function
(a2 — 2 a a' cos. 6 + a72)
BOOK I.] NEWTON'S PRINCIPIA. 117
and develope it according to the cosine of the angle Q and its multiples.
If we make — ; = a, it will become
'
a
— a * — i
a . { ] — 2 a cos. A -j- a *}
Let
( 1 — 2 a cos. 4 + a 2) ~3 = £ b ^ + b C1> cos. tf + b to cos. 2 *
S 6 S
+ b ® cos. 3 0 + &c.
S
b (0), b (l), b (fi), &c. being functions of a and of s. If we take the logarith-
f S S
mic differences of the two members of this equation, relative to the vari
able d} we shall have
1 — 2 a cos. 6 + a. z % b W + b (1) cosJ+b W cos.
S S g
Multiplying this equation crosswise, and comparing similar cosines, we
find generally
(i— 1) (1 + ^b^-D — (i + s— 2)ab<1-2>
b G) = - s-^ - - - -^ -
. a
We shall thus have b (2>, b ®\ &c. when b W and b P) are known.
S B
If we change s into s + 1, in the preceding expression of (1 — 2 a cos. &
~ s
a 2) , we shall have
(1— 2«cos. d+uz} "zr^bW + bWcos. 0+b ® cos.2
8 + 1 S + l 8 + 1 B + l
Multiplying the two members of this equation, by 1 — 2 « cos. rf + «%
and substituting for ( 1 — 2 a cos. 0 + a !) ~ ' its value in series, we shall
have
£ b (c> + b (1> cos. d + b & cos. 2 0 + &c.
S3 S
= (1 — 2acos.0+a2)£ b® + b(1>cos.0 + b^cos. 20 + &C.J
S + l S+ 1 8 + 1
whence by comparing homogeneous terms, we derive
b «> = (1 + «2)b (0_«bli-1) — ab^i + 1).
,,,, r . • S+l 8 + 1 S + l
1 he formula (a) gives
i(l + «2)bW — (i + sjob'1-"
s+l _S.a
Tiie preceding expression of b '• will thus become
L+J
— S
H3
118 A COMMENTARY ON [SECT. XL
Changing i into i + 1 in this equation we shall have
_ ___
i — s + 1
and if we substitute for b (i+J) its preceding value, we shall have
s + l
b 0 + D = _ 1±-1 _ — _ i+J
s 0 — s) (l — S + l)a
These two expressions of b (i) and b (i + J) give
s s
l.(i + g.)bc)-2.i-s+ l«bo.*')
S
! _ - _ ! _ • f M
s + l (1— «T
substituting for b (i + J) its value derived from equation (u), we shall have
~8(i^W~ ~; (c)
an expression which may be derived from the preceding by changing i
into — i, and observing that b (i) = b (-i). We shall therefore have by
means of this formula, the values of b (0), b (1), b (2), &c. when those of
s+l s+l s+l
b (% b (l\ b (2), &c. are known.
a as
Let X, for brevity, denote the function 1 — 2 o cos. 6 + a 2. If we
differentiate relatively to a, the equation
X -•• = £ b (°> + b (1) cos. 6 + b ® cos. 2 6 + &c.
88 8
we shall have
d b (°) d b »> d b (2)
— 2 s (a — cos. 6} X ~ • - 1 = A . —'- 1 f — cos. 6 + -j8 — cos. 2 6 + &c.
2 da da da
But we have
— a + cos. ^ =
2 a
We shall, therefore, have
7\. _ Q j ~1 *V\JJ* " -| «JV*-«
a a ^ d a a a
whence generally we get
» _ S (1 ttJ ^ (i) _ _8 ^
da a g + i a
Substituting for b (i> its value given by the formula (b), we shall have
6 + 1
^__i + (i + 2s).« 0) 2(i-s+l)
"dV "" a(] — a2) ' b.W 1 —a2 ' . '+ '
BOOK I.] NEWTON'S PRINCIPIA. 119
If we differentiate this equation, we shall have
2 (i+s)
1 — a
Again differentiating, we shall get
d"
,
'
11
a2/ da
<x(l — a2) ' da2 ' (11— a2)8
d2b(i
4 (i + s) a (3 + a2) 2_il , ni 2(i— s+1)
h 23 ^
_
(1— a2)3 a3/ 1 — a2 da2
,
(1— a2)2 da (I — a2)3
Thus we perceive that in order to determine the values of b and 01
8
its successive differences, it is sufficient to know those of bw and of b(1).
8 8
We shall determine these two as follows :
If we call c the hyperbolic base, we can put the expression of X— s un
der this form
X-8 = (1 — a cflv'— T) -9 (1 — « C — 0V— !)-•.
Developing the second member of this equation relatively to the powers of
c 6 V—1, and c ~ 6 ^~l, it is evident the two exponentials c i e V—1, c — i 6 V— 1
will have the same coefficient which we denote by k. The sum of the
two terms k . c l e v — 1 and kc — i 0 \/ — Ms 2k cos. i 6. This will be the
value of b (i) cos. i 0. We have, therefore, b (i) = 2 k. Again the ex-
8 S
pression of X - s is equal to the product of the two series
sa c« -i + 1 ««c8»V-i + &c.
!• SB
multiplying therefore these two together, we shall have when i = 0
k = l S2a^
and in the case of i = 1,
wherefore
H I
120 A COMMENTARY ON [SECT. XL
b <„ = 2 „
+JO 4
. . . .
That these series may be convergent, we must have a less than unity,
which can always be made so, unless a = a' ; a being = — - , we have only
to take the greater for the denominator.
In the theory of the motion of the bodies «-, /«.', /t", &c. we have occasion
to Ivnovv the values of b (0) and of b ^ when s = % and s = f . In these
8 S
two cases, these values have but little convergency unless « is a small
fraction.
The series converge with greater rapidity when s = — £, and we have
id) f Ll 2 1 1.1.3 4 1.3 1.1.3.5 6 1.3.5 1.1.3 ..7
V K V~2.4a ~4'2A6a ~^6-2A6^a~4^8'273T^10
" 2
In the Theory of the planets and satellites, it will be sufficient to take
the sum of eleven or a dozen first terms, in neglecting the following
terms or more exactly in summing them as a geometric progression whose
common ratio is 1 — « 2. When we shall have thus determined b (0) and
b n\ we shall have b (0) in making i = 0, and s = — £ in the formula (b),
and we shall find
i "
If in the formula (c) we suppose i = I and s = — £ we shall have
, -«
By means of these values of b(0) and of b(1) we shall have by the pre-
i I
ceding forms the values of b (i) and of its partial differences whatever may
sT
be the number i ; and thence we derive the values of b (l) and of its dif
ferences. The values of b (°) and of b (1) may be determined very simply,
BOOK L] NEWTON'S PRINCIPIA. 121
by the following formula?
b w b (»)
b w = ~£ .
Again to get the quantities A <% A 'l\ &c. and their differences, we
must observe that by the preceding No., the series
£ A ») -f- A ») cos. 0 + A U cos. 2 0 + &c.
results from the developement of the function
a cos. 6 _ i
— — -- (a2 — 2 a a' cos. d + a'2) *,
into a series of cosines of the angle 6 and of its multiples. Making ~ = «,
this same function becomes
S
which gives generally
ACi> = _±.b<»;
a i
when i is zero, or greater than 1, abstraction being made of the sign.
In the case of i = ], we have
We have next
/dA«x ± \ ,da.
V da )~ " a' ' da \daJ;
But we have -,— = — ; therefore
da a'
d b w
(iS - _ J_ _i_
a/~ a/z*da
d
and in the case of i = ] , we have
d b »)
/dAWx JLJ i )
V da ;~ a/2 I da J
Finally, we have, in the same case of i = I
d °~ b ti}
d2
/d2 AWx J_ i
V d a2 y ~ a'3' d «z ;
122 A COMMENTARY ON [SECT. XI.
d 3 b «
/d3 A(i\ J __ £
V da3 / a'4' da3 '
&c.
To get the differences of A (i) relative to a', we shall observe that A w
being a homogeneous function in a and a', of the dimension — 1, we
have by the nature of such functions,
/dA«x, ,/dA«x Am
a (—5 - ) + a'( — 1—7-1 = — A (1);
v d a / ' vda'/
whence we get
--__
da' - da
a, d'
A W\ /d A
= 2 A '" + 4
, 3 /d 3 A «x . fo /d A <1K ,d2A 0). /d^A^x .
a ( -j— /T~ ) = — 6A — I8 a I j — ) — 9a ( —\ — F ) — a ( ~i — r ) '
\da/3/ \da/ \da2/ xdaj/
&c.
We shall get B (i) and its differences, by observing that by the No. pre
ceding, the series
•i B(0> + B W cos. 6 + B ^ cos. 2 Q + &c.
is the developement of the function
a'-3 (1 —2 a cos. 6 + a2)"^
according to the cosine of the angle 6 and its multiples. But this function
thus developed is equal to
a'-3 f£b<°> + b«> cos. d + b® cos. 2 6 + &c.)
II 1 I '
therefore we have generally
a
Whence we derive
dbm d'b«
; *
_B^x J_ _|_; /d'BWx ^ |
da / a/4< da V d a2 / ~ a/s ' d a2
Moreover, B (i> being a homogeneous function of a and of a', ot the
dimension — 3 we have
a
d a / V d a
BOOK I.] NEWTON'S PRINCIPIA. 123
whence it is easy to get the partial differences of B w taken relatively to
a' by means of those in a.
In the theory of the Perturbations of /*', by the action of p, the values
of A Ci) and of B (1), are the same as above with the exception of A (il which
in this theory becomes , -- b (1). Thus the estimate of the values of
2i a ±
2
A (i), B ('°, and their differences will serve also for the theories of the two
bodies /«. and fjf.
517. After this digression upon the developement of R into series, let
us resume the differential equations (X'), (Y), (Z') of Nos. 513, 514; and
find by means of them, the values of 3 g, 8 v, and d s true to quantities
of the order of the excentricities and inclinations of orbits.
If in the elliptic orbits, we suppose
f = a(l + u,); e'=a'(l+u/):
v = n t + s + v7; v' = n' t — s' + v/;
we shall have by No. (488)
u, = — e cos. (n t + s — -a}; u/ = — e' cos. (n' t + s' — «/) ;
v, = 2 e sin. (n t + £ — «r) ; v/ = 2 e' sin. (n' t + e' — */) ;
n t + £, n' t + e' being the mean longitudes of /*, fi! ; a, a' being the semi-
axis-majors of their orbits ; e, e' the ratios of the excentricity to the semi-
axis-major ; , and lastly «r, •&' being the longitudes of their perihelions. All
these longitudes may be referred indifferently to the planes of the orbits,
or to a plane which is but very little inclined to the orbits ; since we ne
glect quantities of the order of the squares and products of the excen
tricities and inclinations. Substituting the preceding values in the ex
pression of R in No. 515, we shall have
R = ~ 2 A « cos. i (n' t — n t + i7 — f)
e cos.Ji (n' t — n t + f' — t) + n t + t —
e' cos.{i (n7 t — n t + e' — e) + n t + t — */};
the symbol 2 of finite integrals, extending to all the whole positive and
negative values of i, not omitting the value i = 0.
Hence we obtain
124 A COMMENTARY ON [SECT. XI.
d
+ n t + £ — *'};
the integral sign 2 extending, as in what follows, to all integer positive
and negative values of i, the value i = 0 being alone excepted, because
we have brought from without this symbol, the terms in which i = 0 : /*' g
is a constant added to the integral/" d R. Making therefore
. .
i(n — n') — n
.
r i (n — n') — n I V d
taking then for unity the sum of the masses M + /«<, and observing that
(237) M + ^ = n2, the equation (X') will become
BOOK I.] NEWTON'S PRINCIPIA. 125
+ n2 («.' C e cos. (n t -f i — -a]
+ ri 2 v/ D e' cos. (n t + £ — »')
+ n - fi! 2 C W e cos. £i (n' t — n t + sf — i) + n t + « — wj
+ n V 2 D (i) e' cos.£i (n' t — n t + e' — s) + n t + « — • »'};
and integrating
to
uf I \ d a / « — IM j • / / . / \
~ -£- n 2 2 . ^- 7-5- 5 — cos. i (n' t — n t + «' — «)
2 i 2 (n — n') 2 — n 2
+ «.' f ' e cos. (n t + £ — -a] + &' f / e' sin. (n t + £ — »')
/ \ / / \
/ /
— C . n t . e sin. (n t + £ — w) — D . n t. e' sin. (n t + i — a/)
+ ^ 2 li(n--n}*-n* ' 6/ °°sji (n/ 1 " " n t +i/~ §) +n t + g~w/?'
fx and f/ being two arbitraries. The expression of d £ in terms 6 u, found
in No. 514 will give
_
i^-l1T^^
— //re cos. (n t + £ — ^) — /«/ f ' e' cos. (n t + 2 — ^')
+ £ ,v/ C n t e sin. (n t+ e — ») + £ ^ D n t e' sin. (n t + £ — */)
r,,,
, 2 )J \da/^n — nx ^ _ Cj1 i
•^ I i2(n — ri')2 — n2 Ji (n— n')— n} 2 — ns)
' X e cos. £i (nx t — n t + e' — s) + n t + £ — »}
— ^ - n 2 2 . U(n_n^nr_n2e/ cos. {i(n' t-n t+ e'_«)+n t+£-«r'],
f and f being arbitrary constants independent of f/5 f/.
This value of 5 g, substituted in the formula (Y) of No. 513 will give 3 v
or the perturbations of the planet in longitude. But we must observe that
n t expressing the mean motion of /*, the term proportional to the time,
ought to disappear from the expression of 8 v. This condition determines
the constant (g) and we find
126 A COMMENTARY ON [SECT. XI.
We might have dispensed with introducing into the value of d g the
arbitraries f, f/, for they may be considered as comprised in the elements
e and -a of elliptic motion. But then the expression of 8 v would include
terms depending upon the mean anomaly, and which would not have
been comprised in those which the elliptic motion gives : that is, it is more
commodious to make these terms in the expression of the longitude dis
appear in order to introduce them into the expression of the radius-vector >
we shall thus determine f, and f/ so as to fulfil this condition. Then if we
/d A(i-1}\ . /d A (i-J).\
substitute for a ( — 5 — •; — ) its value — A (l-1) — a ( — -j- — ') , we shall
v d a' / \ d a /
have
/
r — n 2 I
V
A da' ,
d A «>\
i iu> vuiuc r\
/d2 A(0)\
In3! .V
da/
* V da2 /'
Moreover let
,M, 3 n . m , i 2 (n — n7) . {n + i (n — n')] — 3 n 2
. d2
x a
n — n' i2(n_n')2 — n2
•d A «x 2 n , «,) 2 n«E«
f
l
. (i)
a ' 2
n2— {n — i(n — n')}2'
(i _ ]) (2 i_ 1) n a A*-» + (i - 1) n a
G w —
2 [n — i (n — n')J
2 n 2 D (i)
n2— Jn— i (n — n')}2'
BOOK I.] NEWTON'S PRINCIPIA. 127
and we shall have
, d A «x 2n
a -~ aA
2 i*(n — n')f — n2
cos. i (n' t — n t + tr — s)
— ft'fe cos. (n t + «+ «) — /et' f e' cos. (n t + i — »')
+ £p'C.ntesin. (nt + g — w) + £/Dn te'sin. (n t + i — »')
C o a/ 2/dA(l)\ 2n ) -\
. i . 2nsias(-j \-\ iaA®j f . .
»' J n 2 . m . I \ d a / n — n7 I > sin. i
av=£2) rr «aAW + - J-J , TT-T i
2 (_i(n — n')* i (n — n') . U . (n — nx) 2 — n2] )
(n' t — n t + e' — i)
+ (t! . C . n t . e cos. (n t + s — *?} -f- p' D . n t . e' cos. (n t + e — */)
f F ^
I —, « esin. Ji(n't — n t + t' — t) + nt + i—*}~]
n — i (n — n') '
i , ../ \? J ^ *
n — i (n — nO~ ~
tlie integral sign 2 extending in these expressions to all the whole positive
and negative values of i, with the value i = 0 alone excepted.
Here we may observe, that even in the case where the series represent
ed by
2. A (i) cos. 5 (n' t — n t + i' — e)
^\
is but little convergent, these expressions of — and of d v, become con-
a
vergent by the divisors which they acquire. This remark is the more
important, because, did this not take place, it would have been impossible
to express analytically the mutual perturbations of the planets, of whic
the ratios of their distances from the sun are nearly unity.
These expressions may take the following form, which will be useful to
us hereafter. Let
h r= e sin. -a ; h' = e' sin. «/ ;
1 = e cos. w, 1' = e' cos. -a ;
then we shall have
h
— ^ (hf + h'f) cos. (n t + s) — v! (1 f + \' f) sin. (n t + t)
128 A COMMENTARY ON [SECT. XL
+ ^ {1 C + l' D] n t sin. (n t + e) — ' [h C + h'D}n t cos. (n t + 0
2
= ,. _L_. A« + an. . n ,
2 li(n — n')2 1 (n — n') {i 2 . (n _ n') *~— n*} j
sin. i (n' t — n t + t' — s)
' [h C + h' D}. n t . sin. (n t + e)+/*' {1 . C + l' . D} n t. cos. (n t + s)
+ nX.J"-^fin'U(n/t~nt + £/~e) + nt+<} Is
^_hF^+h^cos^ n/t_n ;/_ r
V. n — i(n — iv )
Connecting these expressions of d § and 3 v with the values of § and v
relative to elliptic motion, we shall have the entire values of the radius-
vector of /*, and of its motion in longitude.
518. Now let us consider the motion of p in latitude. For that pur
pose let us resume the formula (Z') of No. 514. If we neglect the pro
duct of the inclinations by the excentricities of the orbits it will become
the expression of R of No. 515 gives, in taking for the fixed plane that
of the primitive orbit of p,
/d R\ (*' z' (*' zf r> t\\ - , i *. , f \
(~dj) ~ IT5""" ^ ' B cos< l ( " n +£ ^ >
the value of i belonging to all whole positive and negative numbers in
cluding also i = 0. Let 7 be the tangent of the inclination of the orbit
of p', to the primitive orbit of ^, and n the longitude of the ascending
node of the first of these orbits upon the second ; we shall have very
nearly
T! = a' 7 sin. (n' t + t' — ll) ;
which gives
= /- . 7. sin. (n' t + E' — n) — ^ • ^ B <l> y sin.(n t+e—n)
-
d z
-. a! S B i1-1) y sin. {i (n' t — n t + s'— s) + n t + s— n]
y
the value here, as in what follows, extending to all whole positive and
negative numbers, i = 0 being alone excepted. The differential equation
BOOK I.] NEWTON'S PRINCIPIA. 129
in 3 of will become, therefore, when the value of (-7- ) is multiplied by
n2 a3, which is equal to unity,
0 = -j-t2- + n 2 a u' — ft' n2. -^ y sin. (n' t + s7 — n)
„/ n 2
H g— a a' B (i> y sin. (n t + g — n)
H «r~ aa/2B P-Dysin. {i (n't — nt + s' — 0 + nt+s — n)] ;
whence by integrating and observing that by 514
8 s = — a 3 u',
s=— - 7 sin. (n' t + •' — n)
n t . y cos. (n t + -e — n)
:-n2-{n-i(n_nOF7Sin^i(n/t"~nt+£/^ + nt + e-n^
To find the latitude of p above a fixed plane a little inclined to that of
its primitive orbit, by naming p the inclination of this orbit to the fixed
plane, and 6 the longitude of its ascending node upon the same plane ; it
will suffice to add to d s the quantity tan. p sin. (v — 0), or tan. p sin. (n t
+ 6 — 0, neglecting the excentricity of the orbit. Call p/ and ^ what p
and 6 become relatively to (*'. If ^ were in motion upon the primitive
orbits of ft', the tangent of its latitude would be tan. p' sin. (n t + s 6'} ;
this tangent would be tan. p sin. (n t + e — 6), if ft continued to move in
its own primitive orbit. The difference of these two tangents is very
nearly the tangent of the latitude of ft, above the plane of its primitive
orbit, supposing it moved upon the primitive orbit of ft'; we have there
fore
tan. p'sin. (n t + s _ ^) _ tan. p sin. (n t + s — 0) = y sin. (n t + e — n).
Let
tan. p sin. 6 = p ; tan. p sin. tf = p' ;
tan. p cos. 6 = q ; tan. p' cos. tf = q' ;
we shall have
y sin. n = p' — p ; y cos. n = q' — q
and consequently if we denote by s the latitude of ft above the fixed plane,
we shall very nearly have
s = q sin. (n t + e) — p cos. (n t + *)
/&' a 2 a'
4 — (p' — p) B « n t sin. (n t + £)
Vor. II j
130 A COMMENTARY ON [SECT. XI.
m-£|^ (q' -q) B» n t cos. (n t + •)
— ^T^' *(q/ ~~ q) ^ (n/ t+l'] ~ (P/ "" P) C°S' (n/ ' + £/)*
519. Now let us recapitulate. Call (g) aud (v) the parts of the radius-
vector and longitude v upon the orbit, which depend upon the elliptic
motion, we shall have
g = (g) + *S'> v = (v) + 5v.
The preceding value of s, will be the latitude of & above the fixed plane.
But it will be more exact to employ, instead of its two first terms, which
are independent of Xj the value of the latitude, which takes place in the
case where p quits not the plane of its primitive orbit. These expressions
contain all the theory of the planets, when we neglect the squares and the
products of the excentricities and inclinations of the orbits, which is in
most cases allowable. They moreover possess the advantage of being
under a very simple form, and which shows the law of their different
terms.
Sometimes we shall have occasion to recur to terms depending on the
squares and products of the excentricities and inclinations, and even to
the superior powers and products. We can find these terms by the pre
ceding analysis, the consideration which renders them necessary will al
ways facilitate their determination. The approximations in which we
must notice them, would introduce new terms which would depend upon
new arguments. They would reproduce again the arguments, which the
preceding approximations afford, but with coefficients still smaller and
smaller, following that law which it is easy to perceive from the deve-
lopement of R into a series, which was given in No. 515 ; an argument
which, in the successive approximations, in found for thejirst time among the
quantities of any order whatever r, and is reproduced only by quantities oj
the orders r + 2, r-f-4, &c.
Hence it follows that the coefficients of the terms of the form
CITI
t . ' . (n t + s), which enter into the expressions of g, v, and s, are ap-
oos«
proximated up to quantities of the third order, that is to say, that the
approximation in which we should have regard to the squares and pro-
BOOK I.] NEWTON'S PRINCIPIA. 131
ducts of the excentricities and inclinations of the orbits would add nothing
to their values ; they have therefore all the exactness that can be desired.
This it is the more essential to observe, because the secular variations of
the orbits depend upon these same coefficients.
The several terms of the perturbations of g, v, s are comprised in the
form
sin
k ' cos. *i (n' t — n t + e' — s) + r n t + r e}9
r being a whole positive number or zero, and k being a function of the
excentricities and inclinations of the orbits of the order r, or of a superior
order. Hence we may judge of what order is a term depending upon a
given angle.
It is evident that the motion of the bodies (*'', (*/", &c. make it neces
sary to add to the preceding values of g>, v, and s, terms analogous to
those which result from the action of y! ; and that neglecting the square of
the perturbing force, the sums of all these terms will give the whole va
lues off, v and s. This follows from the nature of the formulas (X'),
(Y), (Z'), which are linear relatively to quantities depending on the dis
turbing force.
Lastly, we shall have the perturbations of X, produced by the action of
& by changing in the preceding formulas, a, n, h, 1, s, v, p, q, and (i! into
a', n', H', 1', sf, «•', p', q', and (i and reciprocally.
THE SECULAR INEQUALITIES OF THE CELESTIAL MOTIONS.
520. The perturbing forces of elliptical motion introduce into the expres-
d v
sions off, j-- , and s of the preceding Nos. the time t free from the sym
bols sine and cosine, or under the form of arcs of a circle, which by in
creasing indefinitely, must at length render the expressions defective. It
is therefore essential to make these arcs disappear, and to obtain the
functions which produce them by their developement into series. We
have already given, for this purpose, a general method, from which it re
sults that these arcs arise from the variations of elliptic motion, which are
then functions of the time. These variations taking place very slowly
have been denominated Secular Inequalities. Their theory is one of the
most interesting subjects of the system of the world. We now proceed to
expound it to the extent which its importance demands.
1 2
132 A COMMENTARY ON [SECT. XL
By what has preceded we have
•1 — h sin. (n t + «) — 1 cos. (n t + t) — &c.
= a
- U . C + 1'. V] . n t . sin. (n t + g)
/
— ^-{h . C + h' . D} . n t . cos. (n t + i) + y! S
]
d v
-p = n + 2 n h sin. (nt + t) + 2 nl cos. (n t + t) + &c.
Cl t
— ^ {I C + T D] n 2 1 sin. (n t + •)
+ ^ [h C + h' D} n 2 t cos. (n t + i) + / T ;
s = q sin. (n t + e) — p cos. (n t + t) + &c.
— ^- a 2 a' (p — p) B «>. n t . sin. (n t + t)
T*
— ^- a2 a' (q' — q) B <«. n t. cos. (n t + «) + ^ •&'•>
~k
S, T, ^ being periodic functions of the time t. Consider first the expres
sion of -j — , and compare it with the expression of y in 510. The arbi
trary n multiplying the arc t, under the periodic symbols, in the expres
sion of jf- ; we ought then to make use of the following equations found
( 1 L
in No, 510,
0 = X' + 6.X." — Y;
0 = Y' + 6 . Y" + X" — 2 Z ;
Let us see what these X, X', X", Y, &c. become. By comparing the ex
pression of -3 - with that of y cited above, we find
Cl L
X = n + 2 n h sin. (n t + e) + 2 n 1 cos. (n t + s) + fjf T
Y = (i! n2 £hC+h'D} cos. (n t+t) — ^'n * [I C + FD} sin. (nt+i).
If we neglect the product of the partial differences of the constants by
the perturbing masses, which is allowed, since these differences are of the
order of the masses, we shall have by No. 510,
X' = (1^) U + 2 h sin. (n t + 0 + 2 1 cos. (n t + 01
+ 2 n (~) £h cos. (nt + s) — \ sin. (n t + •)}
+ 2 n()sin. (n t + ,) + 2 n()cos. (n t + ,);
X" = 2 n (— ) [h cos. (n t + i) — 1 sin. (n t + OJ
BOOK I.] NEWTON'S PRINCIPIA. 133
The equation 0 = X' + 6 X" — Y will thus become
0 = (^ [I + 2 h sin. (n t + «) + 2 1 cos. (n t + i)J
- - rl 1
sin. (n t + t) + 2 n (^-J cos. (n t + t)
— it/n2 £h C + h'D} cos. (nt + «)4V n2 Jl C+1'DJ sin.(n t + «).
Equating separately to zero, the coefficients of like sines and cosines, we
shall have
If we integrate these equations, and if in their integrals we change 6
into t, we shall have by No. 510, the values of the arbitrages in functions
of t, and we shall be able to efface the circular arcs from the expressions
d v
of -; — and of g. But instead of this change, we can immediately change
01 I
6 into t in these differential equations. The first of the equations shows
us that n is constant, and since the arbitrary a of the expression for g de
pends upon it, by reason of n2 = — 5, a is likewise constant. The two
other equations do not suffice to determine h, 1, e. We shall have a new
d v
equation in observing that the expression of -= — , gives, in integrating,
(-1 t
yn d t for the value of the mean longitude of p. But we have supposed
this longitude equal to n t + s ; we therefore have n t+ « = ,/n d t, which
gives
t 15 + ii - o-
"dt + dt -
and as we have T— = 0, we have in like manner -j— = 0. Thus the two
d t d t
arbitrages n and t are constants ; the arbitraries h, 1, will consequently be
determined by means of the differential equations,
2
13
134 A COMMENTARY ON [SECT. XI.
The consideration of the expression of y^ having enabled us to deter
mine the values of n, a, h, 1, and e, we perceive a priori., that the differen
tial equations between the same quantities, which result from the expres
sion of £, ought to coincide with those preceding. This may easily be
shown a posteriori, by applying to this expression the method of 510.
Now let us consider the expression of s. Comparing it with that of y
citetf above, we shall have
X = q sin. (n t -f- e) — p cos. (n t + ?) + & %
Y = ^ . a* a' B^ (p — p') sin. (n t + «)
+ ^. a2 a' B«> (q — q') cos. (n t + i),
n and t, by what precedes, being constants; we shall have by No. 510,
X" = 0.
The equation 0 = X' + 0 X" — Y hence becomes
0 = ) sin. (n t+ «) - cos. (n t + .)
_ ^-%2 a' B(1> (p — p') sin. (nt+ t)
T?
— £J? a8 a' B'1) (q — q') cos. (n t + 0 ;
TP
whence we derive, by comparing the coefficients of the like sines and co
sines, and changing d into t, in order to obtain directly p and q in
functions of t,
(q.-q'); (3)
= .a*a<Ba>(P-P'); (4)
When we shall have determined p and q by these equations, we shall
substitute them in the preceding expression of s, effacing the terms which
contain circular arcs, and we shall have
s = q sin. (n t + s) — p cos. (n t + t} + p %.
521. The equation ~ = 0, found above, is one of great importance
Cl L
in the theory of the system of the world, inasmuch as it shows that the
mean motions of the celestial bodies and the major-axes of their orbits are
unalterable. But this equation is approximate to quantities of the order
BOOK I.] NEWTON'S PRINCIPIA. 135
p' h inclusively. If quantities of the order (j! h *, and following orders,
produce in -v-^ , a term of the form 2 k t, k being a function of the ele
ments of the orbits of ^ and yJ\ there will thence result in the expression of
v, the term k t2, which by altering the longitude of p, proportionally to
the time, must at length become extremely sensible. We shall then no
longer have
dn
dl '
6ut instead of this equation we shall have by the preceding No.
dn - 2k-
dl '
It is therefore very important to know whether there are terms of the
form k . t 2 in the expression of v. We now demonstrate, that if
we retain only the Jirst power of the perturbing masses, however far may pro-
ceed the approximation, relatively to the powers of the eccentricities and
inclinations of the orbits, the expression v will not contain such terms.
For this object we will resume the formula (X) of No. 513,
acos.y/hdtf sin.v j 2/^R+gf-r— ) r -asin.v/hdt.£cos.v
m V 1 — e2
Let us consider that part of d g which contains the terms multiplied by t 2,
or for the greater generality, the terms which being multiplied by the sine
or cosine of an angle a t + (3, in which a is very small, have at the same
time a2 for a divisor. It is clear that in supposing « = 0, there will re
sult a term multiplied by t 2, so that the second case shall include the first.
The terms which have the divisor a 2, can evidently only result from a
double integration ; they can only therefore be produced by that part of
d g which contains the double integral signyi Examine first the term
2 a cos. vfn d t (? sin. \fd R)
m V (1 — e2)
If we fix the origin of the angle v at the perihelion, we have
1 + e cos. v '
and consequently
a (1 _e2) — P
COS. V = — - " 1 ;
9 f
whence we derive by differentiating,
a n e1)
p z d v . sin. v = — i .dp;
c
14
136 A COMMENTARY ON [SECT. XI.
but we have,
g* d v = d t V m a (1 — e2) = a2. n d t V I — es;
we shall, therefore, have
a n d t g sin. v _ g d g
V 1 — e*~ ~e~
The term
2 a cos, vy n d t . [g sin. vyV R]
m VI — e2
will therefore become
R), or hy,z R _.. d R.
It is evident, this last function, no longer containing double integrals,
there cannot result from it any term having the divisor a 2.
Now let us consider the term
_ 2 a si"' v./n d t [e cos. \fd R]
m V 1 — e*
of the expression of d g. Substituting for cos. v, its preceding value in g,
this term becomes
2 asm, v/n d t. jg — a (1 — e*)} .fd R
me V I — e*
We have
g = aU+ie' + ejfl,
^ being an infinite series of cosines of the angle n t + i, arid of its multi
ples ; we shall therefore have
/E^J {g — a(l -^}}fd R = a/n d t {% e + ^}fd R.
Call %" the integral fyj n d t ; we shall have
a/n d t . If e + %.}fd R = f a e/n d tfd R + a tf'fd R — a//' . d R.
These two last terms not containing a double integral sign, there can
not thence result any term having a* for a divisor; reckoning only terms
of this kind, we shall have
2 a sin, v/n d t {§ cos. vfdE] __ 3 a* e sin, v/n d tfd R
m V I — e2 m V I — ez
n d t ' m
and the radius e will become
.. .
n d t/ m
BOOK L] NEWTON'S PRINCIPIA. 137
(g) and (— -r:) being the expressions of P and of — ~- , relative to the el
liptic motion. Thus, to estimate in the expression of the radius-vector,
that part of the perturbations, which is divided by « 2, it is sufficient to
3 a
augment by the quantity— . x /n d t . fd R, the mean longitude
n t + s, of this expression relative to the elliptic motion.
Let us see how we ought to estimate this part of the perturbations in
the expression of the longitude v. The formula (Y) of No. 516 gives by
substituting ~— . ~-.fn d tfd R for d g and retaining only the terms
111 II (.1 L
divided by a 2,
a 2 n 2 d t * +1J 3a
v = v a "J:1^ __ _ li. -"/n d tfd R;
V 1 — e* m
But we have by what precedes
j . . . , -
d s = - ----- - ; f«d T =r *«n d t V 1 — e2;
VI — e2
whence it is easy to obtain, by substituting for cos. v its preceding value
ing,
2gd2g + dg2
a2n2d t2 "*" d v
V 1 — e*~~ ~ nd t'
in estimating therefore only that part of the perturbations, which has the
divisor a2, the longitude v will become
(v) and (— pt) being the parts of v and -i-p , relative to the elliptic mo
tion. Thus, in order to estimate that part of the perturbations in the ex
pression of the longitude of /A, we ought to follow the same rule which we
have given with regard to the same in the expression of the radius-vector,
that is to say, we must augment in the elliptic expression of the true
longitude, the mean longitude n t + e by the quantity — /n d tfd R.
The constant part of the expression of (~ p.) developed into a series
of cosines of the angle n t + £ and of its multiples, being reduced (see
488) to unity, there thence results, in the expression of the longi-
138 A COMMENTARY ON [SECT. XI.
tude, the term — f n d t / d R. If d R contain a constant term
k y! . n cl t, this term will produce in the expression of the longitude v,
the following one, •=- . k n 2 1 2. To ascertain the existence of such
fd 111
terms in this expression, we must therefore find whether d R contains a
constant term.
When the orbits are but little excentric and little inclined to one ano
ther, we have seen, No. 518, that R can always be developed into an in
finite series of sines and cosines of angles increasing proportionally to the
time. We can represent them generally by the term
k (if . cos. £i' n' t + i n t + A},
i and i' being whole positive or negative numbers or zero. The differen
tial of this term, taken solely relatively to the mean motion of ^, is
— i k . y! . n d t . sin. {V n' t + i n t + A};
this cannot be constant unless we have 0 = i' n' + i n, which supposes
the mean motions of the bodies //, and p' to be parts of one another ; and
since that does not take place in the solar system, we ought thence to con
clude that the value of d R does not contain constant terms, and that in
considering only the first power of the perturbing masses, the mean mo
tions of the heavenly bodies, are uniform, or which comes to the same thing,
^ — = 0. The value of a being connected to n by means of the equation
Cl L
n z = — j , it thence results that if we neglect the periodical quantities, the
major-axes of the orbits are constant.
If the mean motions of the bodies ^ and /u/9 without being exactly com
mensurable, approach, however, very nearly to that condition, there will
exist in the theory of their motions, inequalities of a long period, and
which, by reason of the smallness of the divisor « 2, will become very sen
sible. We shall see hereafter this is the case with regard to Jupiter and-
Saturn. The preceding analysis will give, in a very simple manner, that
part of the perturbations which depend upon this divisor. It hence re
sults that in this case it is sufficient to vary the mean longitude n t + f
3 a
ory*n d t by the quantity — fn d tfd R; or, which is the same, to aug
ment n in the integral,/ n d t by the quantity - ~fd Rj but consider-
BOOK L] NEWTON'S PRINCIPIA.
m
ing the orbit of ^ as a variable ellipse, we have n e = — 3 ; the preceding
variation of n introduces, therefore, in the semi-axis-major a of the orbit,
2 a*fd R
the variation *• .
If we carry the approximation of the value -r— , to quantities of the
vl L
order of the squares of the perturbing masses, we shall find terms propor
tional to the time ; but considering attentively the differential equations of
the motion of the bodies /A, /&', &c. we shall easily perceive that these terms
are at the same time of the order of the squares and products of the ex-
centricities and inclinations of the orbits. Since, however, every thing
which affects the mean motion, may at length become very sensible, we
shall now notice these terms, and perceive that they produce the secular
equations observed in the motion of the moon.
522. Let us resume the equations (1) and (2) of No. 520, and suppose
_(i.' . n . C m ,-Q-JT _ /a'.n.D
they will become
Vl " t r*. •• V 1 fTI VI 1 /
dl
The expression of (0, 1) and of |0, 1| may be very simply determined in
this way. Substituting, instead of C and D, their values determined in
No. 517, we shall have
We have by No. 516,
db«» d2b(0>
d b (0> d s b <°>
and we shall easily obtain, by the same No. —5 — and . 2 in functions
Q CX, tl 06
of b (0) and b (1) ; and these quantities are given in linear functions of b (0)
* i -4
140 A COMMENTARY ON [SECT. XL
and of b (1); this being done, we shall find
~~ 2
3 a2b(1>
fd_A^ ,d'A»\ -I .
' V da ; V da2 ) -2(1 — «*)2'
wherefore
3 ^. n . « 2. b U
cu) = - Mi_.r •
Let
(a2 — 2 a a' cos. 6 + a'z)*= (a, a') + (a, a')' cos. 0+(a, a')'7 cos.
we shall have by No. 516.
(a, a') = £ a', b <°> ; (a, a')' = a', b «>, &c.
We shall, therefore, have
_
Next we have, by 516,
_ 8g'. na*a'. (a, a')'
4 (a'2 — a2)2
d b (1> d2 b W
.-j -- . -j- .
da da2j
2
Substituting for b (1) and its differences, their values in b (0) and b (1), we
I -* -k
shall find the preceding function equal to
Q f(l+a2)b») + J«b»)l
8al -i -U
therefore
+ a2)
2 (1 —a2)2
or
.— — rr 3 /&'. a n£(a2+ a'2) (a, a')' + a a' (a, a'}}
I^JJ - 2 (a' 2 — a 2) 2
We shall, therefore, thus obtain very simple expressions of (0, 1) and
of JO, 1|, and it is easy to perceive from the values in the series of b(0) and
i
~~ e
of b(1), given in the No. 516, that these expressions are positive,- if n is
~ 2
positive, and negative if n is negative.
Call (0, 2) and |0, 2|, what (0, 1) and |0, 1| become, when we change a'
BOOK I.]
NEWTON'S PRINCIPIA.
141
and i/f into a" and &". In like manner let (0, 3), and (0, 3) be what the
same quantities become, when we change a' and pf into a!" and /«/" ; and
so on. Moreover let h", 1" ; h"', 1'", &c. denote the values of h and 1
relative to the bodies /', ft'", &c. Then, in virtue of the united actions of
the different bodies /*', /<//', p'"9 &c. upon p, we shall have
^i ={(o, i) + (o, 2) + (o, 3) + &c.ji — [oTi|.r — [M.!" — &c. ;
(.1 L
_,
Cl L
It is evident that ',
; &c. will be determined by
expressions similar to those of T— - and of ^; and they are easily obtain
ed by changing successively what is relative to & into that which relates
to //,', /«/', &c. and reciprocally. Let therefore
(1,0), IM)]; (1,2), O; &c.
be what
; &c.
(0,1), JOTT); (0,2), |
become, when we change that which is relative to «, into what is relative
to p and reciprocally. Let moreover
(2,0), gof; (2>!)> 153? &c-
be what
(0,2), IM; (0, 1), |M ; &c.
become, when we change what is relative to ^ into what is relative to /«."
and reciprocally; and so on. The preceding differential equations re
ferred successively to the bodies /«-, /"-', ,<*", &c. will give for determining
h, 1, h', 1', h", 1", &c. the following system of equations,
— = {(0, 1) + (0, 2) + &c.] 1 — |0, 1|. 1' — [0, 2| 1" — &c.
£-! = -J(0,l)+(0,2) + &c.]h-
|0,2|b"+&c.
dh
£ = f(l, 0) + (1, 2) + &c.$l' — 11, Oj. 1 — [M 1" — &c.
11' =—{(1,0) + (l,2) + &c.lh'+[I7o|.h +
Cl L
1 h- = {(2, 0) + (2, 1) + &C.J1" — g]
- |M. T - &c.
dlr/
dt
&c.
= —{(-2, 0) + (2, 1) + &c.} . h" + [2,01 h + 12, 1| h' + &c.
(A)
142 A COMMENTARY ON [SECT. XI.
The quantities (0, 1) and (1, 0), |0, 1) and |I, 0| have remarkable rela
tions, which facilitate the operations, and will be useful hereafter. By
what precedes we have
(0, 1) = - . /rg jpy-j .
If in this expression of (0, 1) we change /// into /w, n into n', a into a'
and reciprocally, we shall have the expression of (1, 0), which will con
sequently be
, - SAt.n'a'8. a (a/ a/
4 (a'2 — a2)2 ;
but we have (a, a'/ = (a', a)', since both these quantities result from th
developement of the function (a 2 — 2 a a' cos 6 + a' 2) s into a series or
dered according to the cosine of 6 and of its multiples. We shall, there
fore, have
(0, 1). ^ n' a' = (1, 0). ft.', n a.
But, neglecting the masses /a, /«,', &c. in comparison ,with M, we have
£ M /2 M
n2 = —. -,\ rr2 = -j-.\ &c.
a3 a'3
Therefore
(0, 1) ft V a = (I, 0) ft! V a';
an equation from which we easily derive (1, 0) when (0, 1) is determined.
In the same manner we shall find,
|0, 1| ft V a = |J70| tf V a'.
These two equations will also subsist in the case where n and n' have
different signs ; that is to say, if the two bodies /*, /*' circulated in different
directions ; but then we must give the sign of n to the radical V a, and
the sign of n' to the radical V a'.
From the two preceding equations evidently result these
(0, 2) fj, V a = (2, 0) ft" V a."-, \0^2\ ft V a = [2J"o|. p" V a"', &c.
(I, 2) (if V a'= (2, 1) (jJ1 V a"; [\^2\ p V a = gjj. p" V a"; &c.
523. To integrate the equations (A) of the preceding No., we shall
make
h = N. sin. (g t + /3) ; 1 = N . cos. (g t + /3) ;
h' = N'. sin. (g t + /3) ; 1' = N' cos. (g t + /3) ;
&c.
Then substituting these values in the equations (A), we shall have
N g ={(0, 1) + (0, 2) + &c.JN — IM]. N' — "
N'g=KM) + 0» 2) + &c.]N — jTToJ. N —
N"g = J(2, 0) + (2, 1) + £c.}N"— [270|. N —
Ml N' — &cj
BOOK I.] NEWTON'S PRINCIPIA. 143
If we suppose the number of the bodies /£, //, X'> &c. equal to i ; these
equations will be in number i, and eliminating from them the constants
N, N', &c., we shall have a final equation in g, of the degree i, which we
easily obtain as follows :
Let <p be the function
N2. ^ V a {g — (0, 1) — (0, 2) — &c.}
+ N' V V a'{g — (I, 0) — (1, 2) — &c.}
+ &c.
+ 2 N ft V a i|o7Tj N' + |OT2| N" + &c.J
+ 2 N>' V a l|TT2[N" + jl73| N'"+ &c.}
+ 2 N>" V a" J|2[3] N'" + &c.}
+ &c.
The equations (B) are reducible from the relations given in the pre
ceding No. to these
- o- &c
'
Considering therefore, N, N', N", &c. as so many variables, <f> will be
a maximum. Moreover, <p being a homogeneous function of these varia
bles, of the second dimension ; we have
we have, therefore, <p = 0, in virtue of the preceding equations.
Thus we can determine the maximum of the function <p. We shall first
differentiate this function relatively to N, and then substitute in p, for N,
its value derived from the equation (-pcf) = 0, a value which will be a
linear function of the quantities Nr, N", &c. In this manner we shall
have a rational function whole and homogeneous of the second dimension
in terms of Nr, N", &c. : let <p (1) be this function. We shall differentiate
<f> (1) relatively to N', and we shall substitute in <p (1) for Nr its value derived
from the equation -cr = 0 : we shall have a homogeneous function
of the second dimension in N", N"', &c. : let <p (2) be this function. Con
tinuing thus, we shall arrive at a function <p (i ~ J) of the second dimension,
in N Ci ~ ]) and which will consequently be of the form (N (i ~ V)z. k, k being
a function of g and constants. If we equal to zero, the differential of
<p (i~1) taken relatively to N^"1', we shall have k = 0; which will give
an equation in g of the degree i, and whose different roots will give as
many different systems for the indeterminates N, N', N", &c. : the inde-
144 A COMMENTARY ON [SECT. XL
terminate N^-1) will be the arbitrary of each system; and we shall im
mediately obtain, the relation of the other indeterminates N, N', &c. of
the same system, to this one, by means of the preceding equations taken
in an inverse order, viz.,
p<i-3)x
^-3 '
Let g, gi, g2, &c. be the i roots of the equation in g : let N, N', N", &c.
be the system of indeterminates, relative to the rootg: letN/5 N/, N/', &c.
be the system of indeterminates relative to the root gb and so on : by the
known theory of linear differential equations, we shall have
h = N sin. (g t + /3) + N! sin. (g, t + ft) + N2 (g8 t + &) + &c. ;
h' = N' sin. (g t + /3) + N/ sin. (gl t + ft) + N2' (g2 t + &) + &c. ;
h"= N"sin. (g t + /3) + N/'sin. (gl t + ft) + N2"(g.2 t + &) + &c. ;
&c.
ft ftj ft} &c' being arbitrary constants. Changing in these values of
h, h', h", &c. the sines into cosines ; we shall have the values of 1, 1', 1", &c.
These different values contain twice as many arbitraries as there are roots
g, gl5 g2, &c. ; for each system of indeterminates contains an arbitrary,
and moreover, it has i arbitraries /3, ft, /32, &c. ; these values are conse
quently the complete integrals of the equations (A) of the preceding
No.
It is necessary, however, to determine only the constants N, N1} &c. ;
N,' N/, &c. ; ft ft, &c. Observations will not give immediately the con
stants, but they make known at a given epoch, the excentricities e, e', &c.
of the orbits, and the longitudes », ?/, &c. of their perihelions, and conse
quently, the values of h, h', &c., 1, 1', &c. : we shall thus derive the values
of the preceding constants. For that purpose, we shall observe that if
we multiply the first, third, fifth, &c. of the differentia] equations (A) of
the preceding No., respectively by N. /*. V a, N'. /a'. V a', &c. ; we
shall have in virtue of equations (B), and the relations found in the pre
ceding No. between (0, 1) and (1, 0), (0, 2), and (2, 0), &c.
N . ~ ft, V a + N'. i^ m V a' + N". ^~ y!' V a" + &c.
= g {N. 1 . ft. V a + N'. 1'. i*'. V a' + N". 1". ft". V a" + &c.}
If we substitute in this equation for h, h', &c. 1, 1', &c. their preceding
values ; we shall have by comparing the coefficients of the same cosines
0 = N . Nj . ft V a + N'. NI'. & V a' + N". N,". ft". V a" + &c. ;
0 = N . N2 . ft V a + N'. Ng . p V a' + N". N2". ft". V a" + &c.
BOOK L] NEWTON'S PRINCIPLE 145
Again, if we multiply the preceding values of h, h', &c. respectively by
N./tt. V a, W.fif. V a', &c.
we shall have, in virtue of these last equations,
N . ft h . V a + N' (i/. h'. V a' + N". //' h". V a" + &c.
= {N2. p . V a + N". {*'. V a! + N"2. p". V a" + &c.} sin (g t + |8)
In like manner, we have
N . p 1 . V a + N'. ^ 1'. V a' + N". //' 1". V a" + &c.
= £N2. ^ . V a + N'2. ^. V a' + N"2. ^. V a" + &c.J cos. (g t + /3).
By fixing the origin of the time t at the epoch for which the values of
h, 1, h', 1', &c. are supposed known ; the two preceding equations give
tan B = N ' h ** V a + N/' h/ <*'' V a/ + N" h" A*" ^ a" + &c-
~ N . 1 p . V a + N'. 1 V. V of + N". I" //'. V a" + &c. *
This expression of tan. /3 contains no indeterminate ; for although the
constants N, N', N", &c. depend upon the indeterminate N(i~l\ yet, as
their relations to this indeterminate are known by what precedes, it will
disappear from the expression of tan. 'B. Having thus determined /3, we
shall have N (i~ l\ by means of one of the two equations which give tan. /3;
and we thence obtain the system of indeterminates, N, N', N", &c. rela
tive to the root g. Changing, in the preceding expressions, this root into
gi' &25 gsj &c. we shall have the values of the arbitraries relative to each
of these roots.
If we substitute these values in the expressions of h, 1, h', 1', &c. ; we
hence derive the values of the excentricities e, e', &c. of the orbits, and
the longitudes of their perihelions, by means of the equations
e2 = h2 + l2; e/2 = h/2 + 1/2; &c.
h h'
tan. nt = -j- ; tan. «/ = p- ; &c.
we shall thus have
e2 = N2 + Ni2 + N22 + &c. + 2 N N; cos. J(gl — g) t + ft — £]
+ 2 N N2 cos. J(gsr-g) t + &-<3) } + 2 N! N2 cos.{(Sf-gi) t+^-ft} +&c.
This quantity is always less than (N + N! + N2 + &c.) 2, when the
roots g, gl, &c. are all real and unequal, by taking positively the quanti
ties N, NI, &c. In like manner, we shall have
tan * = N sin< (g f + ^) + NI sin- (gi t + ft) + N2 sin. (g2 1 + &) + &c.
N cos. (g t + /3) + N, cos. (gl t + ft) + N2 cos. (g-2 1 + &) + &c.
whence it is easy to get,
tan (w_crt_S)= NI sin- Ugi-g) t + ff.-ffj + N2 sin.
1
146 A COMMENTARY ON [SECT. XL
Whilst the sum NI + N2 + &c. of the coefficients of the cosines of
the denominator, all taken positively, is less than N, tan. (a — g t — 0)
can never become infinite ; the angle » — g t — /3 can never reach the
quarter of the circumference ; so that in this case the true mean motion
of the perihelion is equal to g t.
524. From what has been shown it follows, that the excentricities of
the orbits and the positions of their axis-majors, are subject to considera
ble variations, which at length change the nature of the orbits, and whose
periods depending on the roots g, g1? g2, &c., embrace with regard to the
planets, a great number of ages. We may thus consider the excentrici
ties as variably elliptic, and the motions of the perihelions as not uniform.
These variations are very sensible in the satellites of Jupiter, and we shall
see hereafter, that they explain the singular inequalities, observed in the
motion of the third satellite.
But it is of importance to examine whether the variations of the excen
tricities have limits, and whether the orbits are constantly almost circular.
We know that if the roots of the equation in g are all real and unequal,
the excentricity e of the orbit of p is always less than the sum N + NI
+ N2 + &c- °f the coefficients of the sines of the expression of h taken
positively ; and since the coefficients are supposed very small, the value
of e will always be inconsiderable. By taking notice, therefore, of the
secular variations only, we see that the orbits of the bodies /A, /&', /*", &c.
will only flatten more or less in departing a little from the circular form ;
but the positions of their axis-majors will undergo considerable variations.
These axes will be constantly of the same length, and the mean motions
which depend upon them will always be uniform, as we have seen in No.
521. The preceding results, founded upon the smallness of the excentricity
of the orbits, will subsist without ceasing, and will extend to all ages past
and future ; so that we may affirm that at any time, the orbits of the
planets and satellites have never been nor ever will be very excentric, at
least whilst we only consider their mutual actions. But it would not be
the same if any of the roots g, gl5 g2, &c. were equal or imaginary : the
sines and cosines of the expressions of h, 1, h', 1', &c. corresponding to
these roots, would then change into circular arcs or exponentials, and
since these quantities increase indefinitely with the time, the orbits would
at length become very excentric ; the stability of the planetary system
would then be destroyed, and the results found above would cease to
take place. It is therefore highly important to show that g, gi, gg, &c.
are all real and unequal. This we will now demonstrate in a very simple
BOOK I.] NEWTON'S PRINCIPIA. 147
manner, for the case of nature, in which the bodies ^ ,«.', &", &c. of the
system, all circulate in the same direction.
Let us resume the equations (A) of No. 528. If we multiply the first
by p . V a . h ; the second, by /A . V a . 1 ; the third by /u/. V a', h' ; the
fourth by /*'. V a'. 1', &c. and afterwards add the results together ; the
coefficients of h 1, h' 1', h" 1", &c. will be nothing in this sum, the coeffi
cients of h' 1 — h 1' will be |07T|. p . V a — flTO). ^'. V a', and this will
be nothing in virtue of the equation |0, 1|. ft. V a = |1, 0|. /«/. V a' found
in No. 522. The coefficients of h" 1 — h 1", h" 1' — h' 1", &c. will be
nothing for the same reason ; the sum of the equations (A) thus prepared
will therefore be reduced to
hdh + ldl , h'dh' + l'dl' , . , ,
^ .p.V a + gi .(*'. Va'+ &c. = 0;
and consequently to
0 = e d e . ft, . V a + e' d e'. p!. V a' + &c.
Integrating this equation and observing that (No. 52-1) the semi-axis-
majors are constant, we shall have
e z. (t V a + e' 2. //. V a' + e" 2. fil'. V a" + &c. = constant ; (a)
The bodies /A, ^', fj/'9 &c. however being supposed to circulate in the
same direction, the radicals V a, V a', V a", &c. ought to be taken po
sitively in the preceding equation, as we have seen in No. 522; all the
terms of the first member of this equation are therefore positive, and con
sequently, each of them is less than the constant of the second member.
But by supposing at any epoch the excentricities to be very small, this
constant will be very small ; each of the terms of the equation will, there
fore, remain always very small and cannot increase indefinitely ; the orbits
will always be very nearly circular.
The case which we have thus examined, is that of the planets and
satellites of the solar system ; since all these bodies circulate in the same
direction, and at the present epoch their orbits have little excentricity.
That no doubt may exist as to a result so important, we shall observe
that if the equation which determines g, contained imaginary roots, some
of the sines and cosines of the expressions of h, 1, h', 1', &c. would trans
form into exponentials ; thus the expression of h would contain a finite
number of terms of the form P . cft, c being the number of which the
hyperbolic logarithm is unity, and P being a real quantity, because h or
e sin. w is a real quantity. Let
Q.cf<,P'.cf<, Q'.cf<,P".cf',&c.
be the corresponding terms of 1, h', 1', h", &c. ; Q, P', Q', P", &c. being
K2
H8 A COMMENTARY ON [SECT. XI.
also real quantities : the expression of e 2 will contain the term (P 2 + Q c)
czft; the expression of e' 2 will contain the term (P' 2 + Q' 2) c 2 f c, and
so on ; the first member of the equation (u) will therefore contain the
term
Ifv therefore, we suppose c f c to be the greatest of the exponentials
which contain h, 1, h', 1', &c. that is to say, that in which f is the most
considerable, c2ft will be the greatest of the exponentials which contain
the first member of the preceding equation : the preceding term cannot
therefore be destroyed by any other term of this first member ; so that for
this member to be reduced to a constant, the coefficient of c2ft must be
nothing, which gives
0 =(P2+Q'~)^ Va + (P'2+Q'2)//v'a' + (P"2 + Q"V'Va" + &c.
When V a, V a', V a", &c. have the same sign, or which is tantamount,
when the bodies /ct, /«,', /«,", &c. circulate in the same direction, this equa
tion is impossible, provided we do not suppose P = 0, Q = 0, P' = 0, &c.;
whence it follows that the quantities h, 1, h' ]', &c. do not contain expo
nentials, and that the equation in g does not contain imaginary roots.
If this equation had equal roots, the expressions of h, 1, h', 1', &c. would
contain as we know, circular arcs and in the expression of h, we should
have a finite number of terms of the form P t r. Let Q t r, P' t r, Q' t r, &c.
be the corresponding terms of 1, h', 1', &c. P, Q, P', Q', &c. being real
quantities; the first member of the equation (u) will contain the term
{(Pz + Q°~)f*V a + (P'2+Q/2) /a'V a' + (P"2 + Q"2) p." V a" + &c.}. t2r.
If tr is the highest power of t, contained by the values of h, 1, h' V, &c. ;
t 2 r will be the highest power of t contained in the first member of the
equation (u) ; thus, that this member may be reduced to a constant, we
must have
0 = (P2+Q2)/* Va + (P/2+ Q' ')/<*' Va' + &c.
which gives
P = 0, Q = 0, P' = 0, Q' = 0, &c.
The expressions of h, 1, h', T, &c. contain therefore, neither exponen
tials nor circular arcs, and consequently all the roots of the equation in g
are real and unequal.
The system of the orbits of /«., ///, /«/', &c. is therefore perfectly stable
relatively to their excentricities ; these orbits merely oscillate about a
mean state of ellipticity, which they depart from but little by preserving
the same major-axis : their excentricities are always subject to this condi-
BOOK I.]
NEWTON'S PR1NCIPIA.
149
tion, viz. that the sum of their squares multiplied respectively by the masses
of the bodies and by the square roots of the major-axes is always the same.
525. When we shall have determined, by what precedes, the values of
e and of » ; we shall substitute in all the terms of the expressions of f;
and -T- , given in the preceding Nos., effacing the terms which contain
Cl I
the time t without the symbols sine and cosine. The elliptic part of these
expressions will be the same as in the case of an orbit not disturbed, with
this only difference, that the excentricity and the position of the perihe
lion are variable ; but the periods of these variations being very long, by
reason of the smallness of the masses ^, v>', /*", &c. relatively to M ; we
may suppose these variations proportional to the time, during a great
interval, which, for the planets, may extend to many ages before and
after the given epoch.
It is useful, for astronomical purposes, to obtain under this form, the
secular variations of the excentricities and perihelions of the orbits : we
may easily get them from the preceding formulae. In fact, the equation
e 2 = h * + 1 2, gives ederrhdh+ldl; but in considering only the
action of /«/, we have by No. 522,
wherefore
h I'J;
but we have h' 1 — h 1' = e e' sin. («/ — ^) ; we, therefore, have
.) ;
. e sn.
thus, with regard to the reciprocal action of the different bodies /•*', ,«.", &c.
we shall have
- |oTl|. e' sin. (~' — «) + [072|. e" sin. (*" — »•) + &
^ - . —
&c.
d e
- = ]I70J e sin. (» — »') + |1, 2| e" sin. («" — »') + &c.
t "~ •^— — •
d t
&c.
e sn.
2~J e' sin.
&c.
K3
150 A COMMENTARY ON [SECT. XL
The equation tan. « = y , gives by differentiating
e 2 d w = 1 d h — hdl.
With respect only to the action of p,', by substituting for d h and d 1
their values, we shall have
= (0, 1) (h* + P) _ JOTTJ. {h h' + 1 1'};
which gives
^ = (0, 1) — |0, 1[. X COS. (J — w);
we shall, therefore, have, through the reciprocal actions of the bodies
ft, (jf, fjf't &C.
•^ =(0,l) + (0,2) + &c.— (OH]. — cos.(w'— «•)— |(jr2].— cos.(w"— •*}— &c.
d^=(
f^i<
&c.
If we multiply these values of -r- , -5— , &c. T-— , T-— , &c. by the time t ;
we shall have the differential expressions of the secular variations of the
excentricities and of the perihelions, and these expressions which are only
rigorous whilst t is indefinitely small, will however serve for a long in
terval relatively to the planets. Their comparison with precise and distant
observations, affords the most exact mode of determining the masses of the
planets which have no satellites. For any time t we have the excentricity
e, equal to
de t2 dze
e, -i— , -T — i , &c. being relative to the origin of the time t or to the given
Cl t Cl L
d e
epoch. The preceding value of -5 — will give, by differentiating it, and
d * e d 3 e
observing that a, a', &c. are constant, the values of -7—5 , -7-7-3) &c. ; we
Cl L Cl I
can, therefore, thus continue as far as we wish, the preceding series, and
by the same process, the series also relative to -a : but relatively to the
planets, it will be sufficient, in comparing the most ancient observations
BOOK I.]
NEWTON'S PRINCIPIA.
151
which have come down to us, to take into account the square of the time,
in the expressions of the series of e, e', &c. -a, «/, &c.
526. We will now consider the equations relative to the position of the
orbits. For this purpose let us resume the equations (3) and (4) of
No. 520,
-
By No. 5 16, we have
a2 a'.
and by the same No.,
We shall therefore have
= a.
Sb«>
£\ 2 '
3 tif . n . a 2 b (i)
4 4 (1 — a2)2 '
The second member of this equation is what we have denoted by (0, 1)
in 522 ; we shall hence have
^| = (0, 1) (q' - q) ;
^ = (0,l)(p-p');
Hence, it is easy to conclude that the values of q, p, q', p', &c. will be
determined by the following system of differential equations :
j-3 = J(0, 1) + (0,2) +&c.}. p — (0, l)p'— (0, 2)p" — &c.
^=-uo,
= {(l> 0)
-$-£=: — £(2;0) + (2,
&c.
' 2)+&c.} . q + (0, 1) q' + (0, 2) q" + &c.
, 2)+&c.J . q' + (1, 0) q + (1, 2) q" + &c.
+ (2,1) +&c.}.p"— (2,0)p — (2, l)p'_&c.
\ -q" +(2,0)q+(2, l)q' + &c.
K 4
(0
153 A COMMENTARY ON [SECT. XI.
This system of equations is similar to that of the equations (A) of No.
522: it would entirely coincide with it, if in the equations of (A) we were
to change h, 1, h', F, &c. into q, p, q', p', &c. and if we were to suppose
|OTT| = (0,1);
|l»Q| = (1,0);
&c.
Hence, the process which we have used in No. 528 to integrate the
equations (A) applies also to the equations (C). We shall therefore
suppose
q =N Gos.(gt+/3) + NlCos. (git+/S,) + N8 cos. (
p =N sin. (gt+/3) + N! sin. (glt + /31) + N2 sin. (
q' = N'cos. (gt+/3) + N/cos. (git+ft)+N2'cos.(g2t+/32)+&c.
p' = N'sin. (gt+^ + N/sin. (gl t+ft) + N2'sin. (gat+&) + &c.
&c.
and by No. 523, we shall have an equation in g of the degree i, and whose
different roots will be g, gl9 g2, &c. It is easy to perceive that one of
these roots is nothing; for it is clear we satisfy the equations (C) by sup
posing p, p', p", &c. equal and constant, as also q, q', q", &c. This
requires one of the roots of the equation in g to be zero, and we can
thence depress the equation to the degree i — ]. The arbitraries
N, Nj, N', &c. /3, /315 &c. will be determined by the method exposed in
No. 523. Finally, we shall find by the process employed in No. 524.
const. = (p2 + q2) p V a + (p/2 + q'2) tf V a' + &c.
Whence we conclude, as in the No. cited, that the expressions of p, q,
p', q7, &c. contain neither circular arcs nor exponentials, when the bodies
p, yJ> p", &c. circulate in the same direction : and that therefore the equa
tion in g has all its roots real and unequal.
We may obtain two other integrals of the equations (C). In fact, if
we multiply the first of these equations by /M V a, the third by /// V a',
the fifth by //' V a", &c. we shall have, because of the relations found in
No. 522,
° = 3~t " V a + Tt' 'a/ V H/ + &c>;
which by integration gives
constant = q ^ V a + q' /".' V of + &c (1)
In the same manner we find
constant = p ^ V a + p' /// V of + &c. . . . . (2)
Call <p the inclinatior of the orbit of p to the fixed plane, and 6 the Ion-
BOOK I.] NEWTON'S PRINCIPIA. 153
gitude of the ascending node of this orbit upon the same plane ; the lati
tude of i* will be very nearly tan. <p sin. (n t -f £ — ^): Comparing this
value with q sin. (n t + t) — p cos. (n t + 0» we shall have
p = tan. <p sin. d ; q = tan. <p cos. d ;
whence we obtain
tan. <p = V (p 2 + q 2) ; tan. d = ¥- ;
We shall, therefore, have the inclination of the orbit of //, and tne po
sition of its node, by means of the values of p and q. By marking suc
cessively with one dash, two dashes, &c. relatively to /M/, /«•", &c. the values
of tan. <p, tan. 0, we shall have the inclinations of the orbits of // p", &c.
and the positions of their nodes by means of p', q', p", q", &c.
The quantity V p 2 + q 2 is less than the sum N -j- Nj + N2 + &c. of
the coefficients of the sines in the expression of q ; thus, the coefficients
being very small since the orbit is supposed but little inclined to the fixed
plane, its inclination will always be inconsiderable ; whence it follows, that
the system of orbits is also stable, relatively to their inclinations as also to
their excentricities. We may therefore consider the inclinations of the
orbits, as variable quantities comprised within determinate limits, and the
motion of the nodes as not uniform. These variations are very sensible
in the satellites of Jupiter, and we shall see hereafter, that they explain
the singular phenomena observed in the inclination of the orbit of the
fourth satellite.
From the preceding expressions of p and q results this theorem :
Let us imagine a circle whose inclination to a fixed plane is N, and of
•which the longitude of the ascending node is g t + ft ; a^so ^ us imagine
upon this first circle, a second circle inclined by the angle NI , the longitude
of whose intersection with the former circle is gi t + ft ; upon this second
circle let there be a third inclined to it by the angle N2 , the longitude of
whose intersection with the second circle is g2 t + j32 , and so on ; the po
sition of the last circle will be that of the orbit of p.
Applying the same construction to the expressions of h and 1 of No.
523, we see that the tangent of the inclination of the last circle upon the
fixed plane, is equal to the excentricity of /*'s orbit, and that the longitude
of the intersection of this circle with the same plane, is equal to that of
the perihelion of /t's orbit.
527. It is useful for astronomical purposes, to have the differential va
riations of the nodes and inclinations of the orbits. For this purpose, let
us resume the equations of the preceding No.
154 A COMMENTARY ON [SECT. XL
tan. ? = V (p2 + q2), tan. 6 = £- .
Differentiating these, we shall have
d if) = d p sin. 6 + d q cos. 6 ;
, _ d p cos. 6 — d q sin. 6
tan. p
If we substitute for d p and d q, their values given by the equations (C)
of the preceding No. we shall have
j= (0, 1) tan. <f> sin. (6 — ff) + (0, 2) tan p" . sin. (d —
^=_ {(0, l)+(0,2) + &c.J+(0, 1) cos. (, _ ,
In like manner, we shall have
-^=(1, 0) tan. <p sin. (6f — f) + (\, 2) tan. 9" sin. (* — 0"
(1 L
&c.
Astronomers refer the celestial motions to the moveable orbit of the
earth ; it is in fact from the plane of this orbit that we observe them ; it is
therefore important to know the variations of the nodes and the inclina
tions of the orbits, relatively to the orbit of one of the bodies /*, p', /A", &c.
for example to the orbit of /z. It is clear that
q sin. (n t + ?} — p cos. (n' t + f)
would be the latitude of ft,' above the fixed plane if it were in motion upon
the orbit of p. The latitude of this moveable plane above the same
plane is
q' sin. (n't + e) — p' cos. (n t + e')»
but the difference of these two latitudes is very nearly the latitude of ftf
above the orbit of p; calling therefore <p/ the inclination, and dj the lon
gitude of the node of /»' upon the orbit of ft, we shall have, by what
precedes,
tan. p/ = V (p'_p)«+ (q — q)2; tan. */ = jr •
If we take for the fixed plane, that of (Ss orbit at a given epoch ; we
BOOK I.] NEWTON'S PRINCIPIA. 155
shall have at that epoch p = 0, q = 0 ; but the differentials d p and d q
will not be zero ; thus we shall have.
d p; = (dp' — dp) sin. 8 + (d q' — d q) cos. ff ;
' — d P' — d p) cos. 8 — (d q' — d q) sin. &
tan. <p
Substituting for d p, d q, d p', d q', &c. their values given by the equa
tions (C) of the preceding No., we shall have
ijjL = £(1, 2) — (0, 2)} tan. p" sin. (ff — 6")
+ {(i9 3) __ (0, 3)} tan. ?'" sin. (ff — *"') + &c.
-£ = — f (1, 0) + (1, 2) + (1, 3) + &c.J — (0, 1)
-j. £(i, g) — (0, 2)] . ~ '-^— cos. (ff — ff')
+ {(I, 3) — (0, 3)] . -^ Bl_£_ cos. ff — ff") + &c.
It is easy to obtain from these expressions the variations of the nodes,
and inclinations of the orbits of the other bodies (*"9 ^'"^ &c. upon the
moveable orbit of p.
528. The integrals found above, of the differential equations which deter
mine the variations of the elements of the orbits, are only approximate, and
the relations which they give among the elements, only take place on the
supposition that the excentricities of the orbits and their inclinations are
very small. But the integrals (4), (5), (6), (7), which are given in No.
471, give the same relations, whatever may be the excentricities and in-
x d v ~— "~ v d x .
clinations. For this, we shall observe that —. — * is double the
d t
area described during the instant d t, by the projection of the radius-
vector of the planet fj> upon the plane of x, y. In the elliptic motion, if
we neglect the mass of the planet as nothing compared with that of the
sun, taken for unity, we shall have, by the Nos. 157, 237, relatively to the
plane of p's orbit,
.
In order to refer the area upon the orbit to the fixed plane, we must
multiply by the cosine of the inclination f of the orbit to this plane ; we
shall, therefore, have, with reference to this plane,
— e's)
xdy — ydx . — ^ - ^ /»•(
- J / - = cos. <p V a (1 — e2) = . / •= — ^
d t v <V 1 +
tan.
15-6 A COMMENTARY ON [SECT. XI.
In like manner
x'dy' — y'dx' _ la' (I — e'2) .
d t = V 1 + tan. 2 p' '
&c.
These values of x d y — y d x, x' d y' — y' d x', &c. may be used,
abstraction being made of the inequalities of the motion of the planets,
provided we consider the elements e, e', &c. <p, <pf, Sec. as variables, in
virtue of the secular inequalities; the equation (4) of No. 471 will there
fore give in that case,
a (I — e2) , , /a'(l — e/2) ,
C = ^Vl + tan.'P + * */l + tan.V + &C'
j(x'-.x)(dy'-dy)-(y'-y)d*'-clx)l
'" *** \ d t J '
Neglecting this last term, which always remains of the order ^ p', we
shall have
a'(l — e' 2)
c =
Thus, whatever may be the changes which the lapse of time produces
in the values of e, e', &c. <p, <p', &c. by reason of the secular variations,
these values ought always to satisfy the preceding equation.
If we neglect the small quantities of the order e4, or e2 p5, this equa
tion will give
c = (j, V a + A*' V a' + &c. — | ft, V a {c 2 + tan. * p]
— £ A*' V a' fe' 2 + tan 2 p7} — &c. ;
and consequently, if we neglect the squares of e, e', p, &c. we shall have
P V a -\- (*' V a' + &c. constant. We have seen above, that if we only
retain the first power of the perturbing force, a, a', &c. will be separately
constant ; the preceding equation will therefore give, neglecting small
quantities of the order e 4 or e 2 p 8,
const. = fj. V a {e2 + tan. 2 <p] + /j,' V a {e'2 + tan. 2 <p\ + &c.
On the supposition that the orbits are nearly circular, and but little
inclined to one another, the secular variations are determined (No. 522)
by means of differential equations independent of the inclinations, and
which consequently are the same as though the orbits were in one plane.
But in this hypothesis we have
p = 0, ®' = 0, &c.
the preceding equation thus becoming
constant = e 2 /* V a + e' 2 ^ V a' + e" 2 p." V a" + &c.
an equation already given in No. 524.
BOOK L] NEWTON'S PRINCIPIA. 15T
In like manner the secular variations of the inclinations of the orbits,
are (No. 526) determined by means of differential equations, independent
of excentricities, and which consequently are the same as though the or
bits were circular. But in this hypothesis we have e = 0, e' — 0, &c.
Wherefore
const.=/.i \/a . tan 2 £>+ t*'Va.' . tan. 2 ?'+,«," Va" . tan. " <p" + &c.
an equation which has already been given in No. 526.
If we suppose, as in the last No.
p = tan. <p sin. 6 ; q = tan. <p cos. 6 ;
it is easy to prove that, the inclination of the orbit of & to the plane of
x, y being (p, and the longitude of its ascending node reckoned from the
axis of x being 0, the cosine of the inclination of this orbit to the plane of
x, z, will be
q
V ( 1 + tan. 2 p) '
y ~"^ — -
Multiplying this quantity by - — ~" — - , or by its value Va.(l — e2),
Cl L
v f I TJ r ___ y /"J Y
we shall have the value of -- -, - ; the equation (5) of No. 471,
Cl L
will therefore give us, neglecting quantities of the order & 2,
a (1 — e2) , , /a'. (1 — e/2)
C =
We shall find, in like manner, that the equation (6) of No. 471, gives
If in these two equations we neglect quantities of the order e* or es <f> ;
they will become
const. = i* q . V a + pf q' V a' + &c.
const. = ft p V a + // pr V a' + &c.
equations already found in No. 526.
Finally, the equation (7) of No. 471, will give, observing that by 478,
m_ _ 2 m __ d x8 + dy2 + dz2
V ' g d tz
and neglecting quantities of the order p (*',
const. = £ + ^ + ^ + &c.
These duTerent equations subsist, when we regard inequalities due to
very long periods, which affect the elements of the orbits of ^ p', &c.
We have observed in No. 521, that the relation of the mean motions of
these bodies may introduce into the expressions of the axis-majors of the
158 A COMMENTARY ON [SECT. XI.
orbits considered variable, inequalities whose arguments proportional to
the time increase very slowly, and which having for divisors the coeffi"
cients of the time t, in these arguments, may become sensible. But it is
evident that, retaining the terms only which have like divisors, and consi
dering the orbits as ellipses whose elements vary by reason of those terms,
the integrals (4), (5), (6), (7), of No. 471, will always give the relations
between these elements already found; because the terms of the order
/u, (if which have been neglected in these integrals, to obtain the relations,
have not for divisors the very small coefficients above mentioned, or at
least they contain them only when multiplied by a power of the perturb
ing forces superior to that which we are considering.
529. We have observed already, that in the motion of a system of
bodies, there exists an invariable plane, or such as always is of a
parallel situation, which it is easy to find at all times by this condition, that
the sum of the masses of the system, multiplied respectively by the pro
jections of the areas described by the radius-vectors in a given time is a
maximum. It is principally in the theory of the solar system, that the re
search of this plane is important, when viewed with reference to the proper
motions of the stars and of the ecliptic, which make it so difficult to astro
nomers to determine precisely the celestial motions. If we call 7 the
inclination of this invariable plane to that of x, y, and n the longitude of
its ascending node, it is easily found that
c"
tan. /sin. rirr — ; tan. y cos.
and consequently that
u.Va(l — e2) sin. p sin. 0-fVvV '(1 — e/2) sin. p' sin.
tan.y sin. n z= - - * — ==• - - - — - - : -
(\ — e2)cos. p+^'Va'(l — e'2) cos.
_, — e2). sin. pcos. 6-\-(jf V&'(\ — e/2) sin. p' cos.0'+&c.
"7*
(1 — e2) .cos. f + ^V a'(l— e'2) .cos.
We shall determine very easily, by means of these values, the angles 7
and n. We see that to determine the invariable plane we ought to know
the masses of the comets, and the elements of their orbits ; fortunately
these masses appear to be so very small that we may, without sensible
error, neglect their action upon the planets : but time alone can clear up
this point to us. We may observe here, that relatively to this invariable
plane the values of p, q, p', q', &c. contain no constant tei'ms ; for it is
evident by the equations (C) of No. 526, that these terms are the same for
p, p', p", &c. and that they are also the same for q, q', q", &c. ; and since re
latively to the invariable plane, the constants of the first members of the
BOOK L] NEWTON'S PRINCIPIA. 159
equations (1) and (2) of No. 526 are nothing: the constant terms disap
pear, by reason of these equations, from the expressions p, p', &c.
q, q', &c.
Let us consider the motion of the two orbits, supposing them inclined
to one another, by any angle whatever : we shall have by No. 528,
c — sin. <p cos. 6 . p V a ( 1 — e 2) + sin. <f>' . cos. 6' . (jf V af (1 — e' 2) ;
c" = sin. <p sin. 6 . i* V a (1 — e2) +sin. <p' . sin. (f . ,</ V a' (I — e' a).
Let us suppose that the fixed plane to which we refer the motion of the
orbits, is the invariable plane of which we have spoken, and by reference
to which the constants of the first members of these equations, are no
thing, as may easily be shown. The angles <p and <p' being positive, the
preceding equations give the following ones :
p V a (1 — e2) . sin. <p =//V~a' (1 — e/2) . sin. <f>' ;
sin. 6 = — sin. 6' ; cos. 0 = — cos. ^ ;
whence we derive 6' = 6 + the semi circumference ; the nodes of the or
bits are consequently upon the same line ; but the ascending node of the
one coincides with the decending node of the other ; so that the mutual
inclination of the two orbits is equal to <p + <p'.
We have by No. 528,
c = /» V a ( 1 — e2). cos. <p + ft/ V a' ( 1 — e' 2) cos. $' ;
by combining this equation with the preceding one between sin. <p and
sin. p', we shall have
os.p. V a(l _e2)=c2+At2a(l — e2)— i*'*. a'(l— e'2).
If we suppose the orbits circular, or at least having excentricity so small
that we may neglect the squares of their excentricities, the preceding
equation will give p constant : for the same reason <pf will be constant ; the
inclinations of the planes of the orbits to the fixed plane, and to one ano
ther, will therefore be constant, and these three planes will always have a
common intersection. It thence results that the mean instantaneous va
riation of this intersection is always the same ; because it can only be a
function of these inclinations. When they are very small, we shall easily
find by No. 528, and in virtue of the preceding relation between sin. <p
and sin. p', that for the time t, the motion of this intersection is
— {(0,1) + (1,0)}. t.
The position of the invariable plane to which we refer the motion of
the orbits, may easily be determined for any instant whatever ; for we
have only to divide the angle of the mutual inclination of the orbits into
two angles, <p and <f> ', such as that we have in the preceding equation be-
1»30 A COMMENTARY ON [SECT. XI.
tween sin. <p and sin. <pr. Designating, therefore, this mutual inclination
by w, we shall have
// V a! (1 — e/2). sin. «
tan. p = •
a (1 — e2) + iff V a' (1 — e/2) . cos. «
SECOND METHOD OF APPROXIMATION OF THE CELESTIAL MOTIONS.
530. We have already seen that the coordinates of the celestial bodies,
referred to the foci of the principal forces which animate them, are deter
mined by differential equations of the second order. We have integrated
these equations in retaining only the principal forces, and we have shown
that in this case, the orbits are conic sections whose elements are the
arbitrary constants introduced by integration.
The perturbing forces adding only small inequalities to the elliptic mo
tion, it is natural to seek to reduce to the laws of this motion the troubled
motion of the celestial bodies. If we apply to the differential equations
of elliptic motion, augmented by the small terms due to the perturbing
forces, the method exposed in No. 512, we can also consider the celestial
motions in orbits which turn into themselves, as being elliptic; but the
elements of this motion will be variable, and by this method we shall ob
tain their variations. Hence it results that the equations of motion, being
differentials of the second order, not only their finite integrals, but also
their infinitely small integrals of the first order, are the same as in the
case of invariable ellipses ; so that we may differentiate the finite equa
tions of elliptic motion, in treating the elements of this motion as con
stant. It also results from the same method that the differential equa
tions of the first order may be differentiated, by making vary only the
elements of the orbits, and the first differences of the coordinates ; pro
vided that instead of the second differences of these coordinates, we sub
stitute only that part of their values which is due to their perturbing
forces. These results can be derived immediately from the consideration
of elliptic motion.
For that purpose, conceive an ellipse passing through a planet, and
through the element of the curve which it describes, and whose focus is
occupied by the sun. This ellipse is that which the planet would invari
ably describe, if the perturbing forces were to cease to act upon it. Its
elements are constant during the instant d t; but they vary from one
instant to another. Let therefore V = 0, be a finite equation to an in
variable ellipse, V being a function of the rectangular coordinates x, y, z
BOOK I.] NEWTON'S PRINCIPIA. 161
and the parameters c, c', &c. which are functions of the elements of ellip
tic motion. Since, however, this ellipse belongs to the element of the
curve described by the planet during the instant d t ; the equation V = 0
will still hold good for the first and last point of this element, by regard
ing c, c', &c. as constant. We may, therefore, differentiate this equation
once in only supposing x, y, z, to vary, which gives
0= (, — ^ d x + ( j — ) d y + (-p — ) d z; (i)
\d x / \d y / ^d z /
We also see the reason why the finite equations of the invariable el
lipse, may, in the case of the variable ellipse, be differentiated once in
treating the parameters as constant. For the same reason, every differ
ential equation of the first order relative to the invariable ellipse, equally
holds good for the variable ellipse ; for let V = 0 be an equation of this
order, V being a function of x, y, z, -T — , -s-4- , -T— , and the parameters
c, c', &c. It is clear that all these quantities are the same for the varia
ble ellipse as well as for the invariable ellipse, which for the instant d t
coincides with it.
Now if we consider the planet, at the end of the instant d t, or at the
commencement of the following one ; the function V will vary from the
ellipse relative to the instant d t to the consecutive ellipse only by the
variation of the parameters, since the coordinates x, y, z, relative to the
end of the first instant are the same for the two ellipses ; thus the function
V being nothing, we have
This equation may be deduced from the equation V = 0, by making
x, y, z, c, c', &c. vary together ; for if we take the differential equation
(i) from this differential, we shall have the equation (i').
Differentiating the equation (i), we shall have a new equation in d c,
d c', &c. which with the equation (i') will serve to determine the parame
ters c, c', &c. Thus it is that the geometers, who were first occupied in
the theory of celestial perturbations, have determined the variations of
the nodes and the inclinations of the orbits : but we may simplify this
differentiation in the following manner.
Consider generally the differential equation of the first order V7 = 0,
an equation which belongs equally to the variable ellipse, and to the in
variable ellipse which, in the instant d t, coincides with it. In the follow
ing instant, this equation belongs also to the two ellipses, but with this
Vor.. II. L
162 A COMMENTARY ON [SECT. XI.
difference, that c, c', &c. remain the same in the case of the invariable
ellipse, but vary with the variable ellipse. Let .V be what V becomes,
when the ellipse is supposed invariable, and V/ what this same function
becomes in the case of the variable ellipse. It is clear that in order to
have V we must change in V, the coordinates x, y, z, which are rela
tive to the commencement of the first instant d t, in those which are rela
tive to the commencement of the second instant; we must then augment
the first differences d x, d y, d z respectively by the quantities d 2 x, d 2 y»
d 2 z, relative to the invariable ellipse, the element d t of the time, being
supposed constant.
In like manner, to get V/, we must change in V, the coordinates
x, y, z, in those which are relative to the commencement of the second
instant, and which are also the same in the two ellipses ; we must then
augment d x, d y, d z respectively by the quantities d 2 x, d 2 y, d 2 z ; finally,
we must change the parameters c, c', &c. into c + d c, c' + d c' ; &c.
The values of d 2 x, d 2 y, d 2 z are not the same in the two ellipses ;
they are augmented, in the case of the variable ellipse, by the quantities
due to the perturbing forces. We see also that the two functions V"
and V/j differing only in this that in the second the parameters c, c', &c.
increase by d c, d c', &c. ; and the values of d2 x, d2 y, d2 z relative to
the invariable ellipse, are augmented by quantities due to the perturbing
forces. We shall, therefore, form V/ — V", by differentiating V in the
supposition that x, y, z are constant, and that d x, d y, d z, c, c', &c.
are variable, provided that in this differential we substitute for d 2 x, d 2 y,
d2 z, &c. the parts of their values due solely to the disturbing forces.
If, however, in the function V" — V we substitute for d2x, dzy, d2z
their values relative to elliptic motion, we shall have a function of x, y, z,
-: — , -j-^- , -: — , c, c', &c., which in the case of the invariable ellipse, is
d t d t d t
nothing; this function is therefore also nothing in the case of the variable
ellipse. We evidently have in this last case, V/ — V' = 0, since this
equation is the differential of the equation V = 0 : taking it from the
equation V/ — V = 0, we have V/ — V" = 0. Thus, we may, in this
case, differentiate the equation V = 0, supposing d x, d y, d z, c, c', &c.
alone to vary, provided that we substitute for d 2 x, d - y, d2 z, the parts
of their values relative to the disturbing forces. These results are exactly
the same as those which we obtained in No. 512, by considerations purely
analytical ; but as is due to their importance, we shall here again present
them, deduced from the consideration of elliptic motion.
BOOK L]
NEWTON'S PRINCIPIA.
163
531. Let us resume the equations (P) of No. 513,
0 -*!15 J. — * 4
U ~ +
-JT2" -jr
d 2 z m z
= dT2;'~p~
If we suppose R = 0, we shall have the equations of elliptic motion,
which we have integrated in (478)- We have there obtained the seven
following integrals
xdy — vdx
c' ~
dt
x d z — z d
3
X
f y d y . d x
z d z .d x -*
c" —
dt
y d z — z d
y.
0 =
0 =
0 -
dt
f 4- x!m-
dy2 + dz2 \
1 T •*• |
PI f m
dt2 I
da . 1 n \
xz + d zz ^
d t2
x d x . d y
dt2
z d z . d y
f J.\- -
dt2 /
dx2-j-dyz ^
dt2
x d x .d z
dt2
L y d y . d z i
m 2m
dt2 J
d x2+ d y* +
d z2
dt2 '
(P)
These integrals give the arbitrages in functions of their first differences;
they are under a very commodious form for determining the variations of
these arbitraries. The three first integrals give, by differentiatino- them,
and making vary by the preceding No. the parameters c, c/, c", and the
first differences of the coordinates,
, x d 2 y — y d 2 x
d c = J , -•?
d t
dc' =
x d 2 z — z d 2 x
" dT~
, „ y d 2 z — z d 2 v
d c"= * = ^~
dt
Substituting for d2 x, d2 y, d2 z, the parts of their values due to the
perturbing forces, and which by the differential equations (P) are
164 A COMMENTARY ON [SECT. XI.
we shall have
, dR
dc =
d R\ /a R%
We know from 478, 479 that the parameters c, c', c" determine three
elements of the elliptic orbit, viz., the inclination <p of the orbit to the
plane of x, y, and the longitude 6 of its ascending node, by means of the
equations
V (c'2 + c"2) c" -
tan. <p = s — -21 ; tan. 6 = —, ;
and the semi-parameter a (1 — e2) of the ellipse by means of the equa
tion
ma(l — e2) = c2 +c/2 + c"J.
The same equations subsist also in the case of the variable ellipse,
provided we determine c, c', c" by means of the preceding differential
equations. We shall thus have the parameter of the variable ellipse, its
inclination to the fixed plane of x, y and the position of its node.
The three first of the equations (p) have given us in No. (479) the
finite integral
0 = c" x — c' y + c z :
this equation subsists in the case of the troubled ellipse, as also its first
difference
0 = c" d x — c' a y + c d z
taken in considering c, c', c" constant.
If we differentiate the fourth, the fifth and the sixth of the integrals
(p), making only the parameters f, f', f", and the differences d x, d y, d z
vary; if moreover, we substitute then for d 2 x, d 2 y, d 2 z, the quantities
" R\ d 1 2 (ilh — d t2 (—\ we shall have
V>~~at \d v)9 Viz;"
+ (x d y - y d x) () + (z d y - y d z) ',
BOOK I.] NEWTON'S PRINCIPIA. 165
Rxl , f /dR
df = d
+ (X d Z — Z d X) (g^) + (y d Z — Z d y) (gy).
Finally, the seventh of the integrals (p)?1 differentiated in the same
manner, will give the variation of the semi-axis-major a, by means of the
equation
d. ~ = 2dR,
the differential d R being taken relatively to the coordinates x, y, z, alone
of the body /*.
The values of f, P, f" determine the longitude of the projection of the
perihelion of the orbit, upon the fixed plane, and the relation of the ex-
centricity to the semi-axis-major ; for I being the longitude of this projec
tion by (479) we have
p
tan. I = ->-;
and e being the ratio of the excentricity to the semi -axis-major, we have
me = V (f2 + f'2 + f"2)-
This ratio may also be determined by dividing the semi-parameter
a (1 — e2), by the semi-axis-major a : the quotient taken from unity will
give the value of e z.
The integrals (p) have given by elimination (479) the finite integral
0 = m g — • h 2 + f x + f'.y + f" z :
this equation subsists in the case of the troubled ellipse, and it determines
at each instant, the nature of the variable ellipse. We may differentiate
it, considering f, f , f" as constant ; which gives
0 = m d s + f d x + f d y + f" d z.
The semi-axis-major a gives the mean motion of /A, or more exactly,
that which in the troubled orbit, corresponds to the mean motion in the
invariable orbit ; for we have (479) n = a ~ 2 V m ; moreover, if we de
note by <£ the mean motion of /t*, we have in the invariable elliptic orbit
d £ = n d t : this equation equally holds good in the variable ellipse,
since it is a differential of the first order. Differentiating we shall have
d * £ = d n . d t ; but we have
San ,m 3anrfR
d n = — -- . d . - = -- ,
2m a m
therefore
3 a n d t. d R
d * I =
m
L3
J66 A COMMENTARY ON [SECT. XT.
and integrating
£ = - .//a n d t . d R.
m JJ
Finally we have seen in (No. 473) that the integrals (p) are equivalent
to but five distinct integrals, and that they give between the seven para
meters c, c', c", f, f, i" e, the two equations of condition
0 = fc" — f c' + PC;
:
in f* + f/2 + f"2 — m
a c2 + c'2 + c//2
these equations subsist therefore in the case of the variable ellipse provid
ed that the parameters are determined as above.
We can easily verify these statements a posteriori.
We have determined five elements of the variable orbit, viz., its inclin
ation, position of the nodes, its semi-axis-major which gives its mean mo
tion, its excentricity and the position of the perihelion. It remains for us
to find the sixth element of elliptic motion, — that which in the invariable
ellipse corresponds to the position of 11 at a given epoch. For this pur
pose let us resume the expression of d t (473)
dt Vm _ d v(l — e2)*"
af = {1 + ecos. (v — »)}«'
This equation developed into series gives (473)
n d t = d v {] + E(1> cos. (v — ») + E® cos. 2 (v — ») + &c.J,
Integrating this equation on the supposition of e and w being con
stant, we shall have
EC1)
/n d t + e - v + E C1) sin. (v — ») -f -5- sin. 2. (v — ») + &c.
tQ
« being an arbitrary. This integral is relative to the invariable ellipse :
to extend it to the variable ellipse, in making every thing vary even to
the arbitrages, E, e, •& which it contains, its differential must coincide with
the preceding one ; which gives
da = de{ (^e-)sin. (v — w; + * (-^) sin. 2 (v - w) + &c.}
— d» fcEWcos. (v — .) + E®cos.2(v — «) + &c.}
v — ro being the true anomaly of (A measured upon the orbit, and » the
longitude of the perihelion also measured upon the orbit, We have de
termined above, the longitude I of the projection of the perihelion upon
a fixed plane. But by (488) we have, in changing v into -a and v, into I
in the expression of v — [3 of this No.
* — 8 = I — 6 + tan. * $ <p sin. 2(1 — 6} + &c.
BOOK I.] NEWTON'S PRINCIPIA. 167
Supposing next that v, v/5 are zero in this same expression, we have
|8 = 0 + tan. 2 £ <f> sin. 2 6 + &c.
wherefore,
*r = I + tan. 8| p. {sin. 2 0 + sin. 2 (I — 6) + &c.}
which gives
d» = dl. {1 + 2 tan.2 |f cos. 2 (I — 6) + &c.J
+ 2 d 0 tan. 2 £ p {cos. 2 d — cos. 2 (I — 6} + bcc.}
dp tan. $p ^ {s[^ 2 , sin> 2 (I _ 0) + &c.}.
cos. ^ £ p
Thus the values of d I, d 0, and d p being determined by the above, we
shall have that of d v ; whence we shall obtain the value of d «.
It follows from thence that the expressions in series, of the radius-vec
tor, of its projection upon the fixed plane, of the longitude whether re
ferred to the fixed plane or to the orbit, and of the latitude which we
have given in (No. 488) for the case of the invariable ellipse, subsist equal
ly in the case of the troubled ellipse, provided we change n t into/n d t,
and we determine the elements of the variable ellipse by the preceding
formulas. For since the finite equations between g, v, s, x, y, z, and
J n d t, are the same in the two cases, and because the series of No. 488
result from these equations, by analytical operations entirely independent
of the constancy or variability of the elements, it is evident these expres
sions subsist in the case of variable elements.
When the ellipses are very excentric, as is the case with the orbits of
the comets, we must make a slight change in the preceding analysis. The
inclination <p of the orbit to the fixed plane, the longitude 6 of its ascend
ing node, the semi-axis-major a, the semi-parameter a (1 — e2), the ex-
centricity e, and the longitude I of the perihelion upon the fixed plane
may be determined by what precedes. But the values of -a and of d -a
being given in series ordered according to the powers of tan. \ p, in order
to render them convergent, we must choose the fixed plane, so as to make
tan. \ p inconsiderable ; and to effect this most simply is to take, for the
fixed plane, that of the orbit of ^ at a given epoch.
The preceding value of d E is expressed by a series which is convergent
only in the case where the excentricity of the orbit is inconsiderable, we
cannot therefore make use of it in this case. Instead, let us resume the
equation
d t V m d v ( 1 — e 2) $
~f~ = [I + ecos. (v — ~)}2'
168 A COMMENTARY ON [SECT. XL
If we make 1 — e = a, \ve have by (489) in the case of the invariable
ellipse,
T being an arbitrary. To extend this equation to the variable ellipse,
we must differentiate it by making vary T, the semi parameter a ( 1 — e 2),
«, and v. We shall thence obtain a differential equation which will de
termine T, and the finite equations which subsist in the case of the in
variable ellipse, will still hold good in that of the variable ellipse.
532. Let us consider more particularly the variations of the elements
of ft's orbit, in the case of the orbits being of small excentricity and but
little inclined to one another. We have given in No. 515. the manner of
developing R in a series of sines and cosines of the form
(jf k cos. (i' n' t — i n t + A)
k and A being functions of the excentricity and inclinations of the orbits,
the positions of their nodes and perihelions, the longitudes of the bodies
at a given epoch, and the major-axes. When the ellipses are variable
all these quantities must be supposed to vary conformably to what pre
cedes. We must moreover change in the preceding term, the angle
i' n' t — i n t into \' J n' d t — i J n d t, or which is tantamount, into
i' % - i £.
However, by the preceding No., we have
The difference d R being taken relatively to the coordinates x, y, z,
of the body p, we must only make vary, in the term
(t! k cos. (i' £' — i C + A)
of the expression of R developed into a series, what depends upon the
motion of this body ; moreover, R being a finite function of x, y, z, x', y', z'
we may by No. 530, suppose the elements of the orbit constant in the
difference d R ; it suffices therefore to make £ vary in the preceding term,
and since the difference of £ is n d t, we have
i (if. k n d t . sin. (V % — i £ + A)
for the term of d R which corresponds to the preceding term of R. Thus,
with respect to this term only, we have
! "2l////knd t.sin. (i' £' — i £ + A);
m
BOOK I.] NEWTON'S PRINCIPIA. 169
£ - '^-ffa k n2 d t2 sin. (i' £' — i £ + A).
If we neglect the squares and products of the perturbing masses, we
may, in the integrals of the above terms, suppose the elements of elliptic
motion constant. Hence £ becomes n t and £', n' t ; whence we get
1 2 i y! n k
;c- - • — \
m (i' n' — i n)
.., . A N
cos- (i n' t — i n t + A)
3 i // a n2 k . ,./ / . • \\
I = -- r^-, - r-^s sin. (i' n' t — i n t + A).
m (i n' — in)2
Hence we perceive that if i' n' — in is not zero, the quantities a and £
only contain periodic inequalities, retaining only the first power of the
perturbing force ; but i and i' being whole numbers, the equation i n' — in
= 0 cannot subsist when the mean motions of p and (t! are incommen
surable, which is the case with the planets, and which can be admitted
generally, since n and n' being arbitrary constants susceptible of all possi
ble values, their exact relation of number to number is not at all probable.
We are, therefore, conducted to this remarkable result, viz., that the
principal axes of the planets, and their mean motions, are only subject to
periodic inequalities depending on their configuration, and that thus in ne
glecting these quantities, their principal axes are constant and their mean
motions uniform, a result agreeing "with 'what has otherwise been found by
No. 521.
If the mean motions n t and n' t, without being exactly commensurable,
approach very nearly to the ratio i' : i ; the divisor i' n' — in is very"
small, and there may result in £ and £' inequalities, which increasing very
slowly, may give reason for observers to suppose that the mean motions
of the two bodies p, (i! are not uniform. We shall see, in the theory of
Jupiter and Saturn, that this is actually the case with regard to these two
planets : their mean motions are such that twice that of Jupiter is very nearly
equal to five times that of Saturn ; so that 5 n' — 2 n is hardly the sixty-
fourth part of n. The smallness of this divisor, renders very sensible the
term of the expression for £, depending upon the angle 5 n' t — 2 n t,
although it is of the order i' — i, or of the third order, relatively to the
excentricities and inclinations of the orbits, as we have seen in No. 515.
The preceding analysis gives the most sensible part of these inequalities ;
for the variation of the mean longitude depends on two integrations, whilst
the variations of the other elements of elliptic motion depend only on
one integration ; only terms of the expression of the mean longitude can
therefore have the divisor (i' n' — in)2; consequently with regard only
A COMMENTARY ON [SECT. XL
to these terms, which, considering the smallness of the divisor ought to
be the more considerable, it will suffice, in the expressions of the radius-
vector, the longitude and latitude, to derive from these terms, the mean
longitude.
When we have inequalities of this kind, which the action of f' produces
in the mean motion of /*, it is easy thence to get the corresponding ine
qualities which the action of p produces in the mean motion of /*' In
fact, if we have regard only to the mutual action of three bodies M, ^ and
/*'; the formula (7) of (471) gives
const = ,dx- + dy + dZ« dx" + dy"+d*«
ilt2- dt2
_ (ft, d x + ft' d x') 2 + (ft, d y + p' d y')2 + (0 d z + ft,' d z') 2
(M +|t6 + p')*d t2
2 My 2 M
+ z* V(x'-x)2-f(y'— y)M-(z'-z)''
The last of the integrals (p) of the preceding No. gives, by substituting
for — the integral 2fd R,
dx2-f-dy2 + dz2_ 2 (M -f ^)
If we then call R', what R becomes when we consider the action of
upon |tt', we shall have
R, _ y. (x xr + y y; + z zQ _ p
(x«+y« + z«)* V"(?— x)2+(y'-y)2+(z'— z)^
dz^_ 2 (M + ft')
"
dt2
the differential characteristic ^ only belonging to the coordinates of the
i i / cur*.- f dx2 + dy2 + dz2 , d x/2 + d y/2 + d z/2
body /*'. Substituting for -- - — «-*| — - - and - — — ,J „ •- --
U. L Cl t
the values in the equation (a), we shall have
'- const
_ const.
2 (M + ,* + /*') dt2
„ 2 / 2
+ " , — z- 2 g H — -• /2 — /8 ^ .
It is evident that the second member of this equation contains no terms
oi the order of squares and products of the ^ &', which have the divisor
i' n' — in; relative, therefore, only to these terms, we shall have
I -j- f^ J d R' = 0;
BOOK I.] NEWTON'S PRINCIPIA. 171
thus, by only considering the terms which have the divisor (V n' — in) 2,
we shall have
3/yVn'dt.d'R' _ _ p(M + v).afjS Sffa n d t . d R
M + (*' ~ iif (M + /TTn ' ~ M. + P
but we have
Sffandt.dR , _ Bffaf n' d t . d' R'
^ = ~~M + p ; ^ = M + ^
we therefore get
^ (M + ^) a n % + p (M + /*) a' n7 £ = 0.
Again, we have
_ V (M + AQ . _V (M + ^0.
a * a' 2
neglecting therefore /A, /"•', in comparison with M, we shall have
A* V a . £ + fit V a'. £' = 0 ;
or
' v s^' r
Thus the inequalities of £, which have the divisor (i' n7 — in) 2, give
us those of £', which have the same divisor. These inequalities are, as
we see, affected with the contrary sign, if n and n' have the same sign, or
which amounts to the same, if the two bodies /* and (i! circulate in the
same direction; they are, moreover, in a constant ratio; whence it follows
that if they seem to accelerate the mean motion of /u-, they appear to re
tard that of (*>' according to the same law, and the apparent acceleration
of jw, will be to the apparent retardation of /",', as pf V af is to /» V a. The
acceleration of the mean motion of Jupiter and the retardation of that of
Saturn, which the comparison of modern with ancient observations made
known to Halley, being very nearly in this ratio ; it may be concluded
from the preceding theorem, that they are due to the mutual action of the
two planets; and, since it is constant, that this action cannot produce in
the mean motions any alteration independent of the configuration of the
planets, it is very probable that there exists in the theory of Jupiter and
Saturn a great periodic inequality, of a very long period. Next, consider
ing that five times the mean motion of Saturn, minus twice that of Jupi
ter is very nearly equal to nothing, it seems very probable that the phe
nomenon observed by Halley, was due to an inequality depending upon
this argument. The determination of this inequality will verify the con
jecture.
The period of the argument i' n' t — i n t being supposed very long,
172 A COMMENTARY ON [SECT. XI.
the elements of the orbits of /*, and /// undergo, in this interval sensible
variations, which must be taken into account in the double integral
ffa k n2 d t2 sin. (V n' t — i n t + A).
For that purpose we shall give to the function k sin. (i' n' t — i n t + A),
the form
Q sin. (i' n' t — i n t + i' e' — i «) + Q' cos. (i' n' t — i n t + i' i — if)
Q and Q' being functions of the elements of the orbits : thus we shall
have
ffa. k n2 d t2 sin. (i' n' 1 — i n t + A) =
n2 a sin. (V n' t— i n t + iV-i 0 / o 2 d Q' 3d2Q 1
(V n' — i n) 2 * X (iV— in)dt (i'n'— in)sdt» "*" C* J
n2 a cos.(iVt— i n t+i' t— i Q f o, 2 d Q 3 d2Q' 1
(I7 n' — i np * t W (i'n'— in)dt (i'n'— in)2dt£+ C' )'
The terms of these two series decreasing very rapidly, with regard to
the slowness of the secular variations of the elliptic elements, we may, in
each series, stop at the two first terms. Then substituting for the ele
ments of the orbits their values ordered according to the powers of the
tune, and only retaining the first power, the double integral above may
be transformed in one term to the form
(F + E t) sin. (i' n' t— i n t + A + H t).
Relatively to Jupiter and Saturn, this expression may serve for many
ages before and after the instant from which we date the given epoch.
The great inequalities above referred to, become sensible amongst the
terms depending upon the second power of the perturbing forces. In
fact, if in the formula
£ = ^~ff^ k n2. d t2. sin. <i' % — i £ + A),
we substitute for £, g their values
3 i &' a n 2 k . ,.,/..
n t • 777-7 — r-r-z sin. (i' n' t — i n t + A) ;
m(i/n/ — in)z
3 i /» a n2 k /a . ,-, , . • ^
n t 7TJ-—. — =— »./ - • sin. (i' n' t — i n t + A),
1x1(1' n' — in)W a'
there will result among the terms of the order (j,z, the following
9iV2a2n4k2 i // V a' + r> V a . .
— pi — 0,., , — = — a • — 7-h • sm- * (i n' t — i n t + A).
8 m8 (Y n7— i n)4 ^ V a
The value of % contains the corresponding term, which is to the one
preceding in the ratio v> V a : — (if V a', viz.
9iV2a2n4k2,. ,/,,-, / , ft V a . Q/w ,. . A,
8m2(i'n'-in)4^ V a + i> ^ «}• ^F^ «in. 2 (i'n' t-i n t + A).
533. It may happen that the inequalities of the mean motion which are the
BOOK I.] NEWTTON'S PRINCIPIA. 173
most sensible, are only to be found among terms of the order of the
squares of the perturbing masses. If we consider three bodies, /*, AS /*"
circulating around M, the expression of d R relative to terms of this or
der, will contain inequalities of the form
k sin. (i n t — i' n' t + I" n" t + A)
but if we suppose the mean motions n t, n' t, n" t such that in — i' n'
-f- \" n" is an extremely small fraction of n, there will result a very sensible
inequality in the value of £. This inequality may render rigorously equal
to zero, the quantity in — i' n' + i" n", and thus establish an equation of
condition between the mean motions and the mean longitudes of the three
bodies /«-, ,«/, y!'. This very singular case exists in the system of Jupiter's
satellites. We will give the analysis of it.
If we take M for the mass-unit, and neglect ^ /*•', &" in comparison with
it, we shall have
2 _ 1 1 1
= a3' = a7"3' "a77"3'
we have then
d £ = n d t ; d £' = n' d t ; d £" = n" d t ;
wherefore
d2£
3 ida
— _rr, Y\ 3 "
dt
2 a2 '
d*£'
3 ^ da'
dt
2n ' a/2
d2 i"
— n"$ a
2 a"2
dt
We have seen in No. 528, that if we neglect the squares of the excen-
tricities and inclinations of the orbits, we have
const. = ii V a + (*'. V a' + y!' V a" ;
which gives
0 = p — + ' j!-a/ + " d a//
V a V a' V a" '
From these several equations, it is easy to get
d2£ _3 $ (U
d t 2 ' n ' a °~
d2T 3 p. n'% n — n"da
d t 2 ' (j!. n ' n' — - n" ' a8 '
i ' £" , _3 m.n"* n — n' da
d t 2 * /*". n n' — n" ' a 2
174 A COMMENTARY ON
Finally the equation
[SECT. XL
R
of No. 531, gives
We have therefore only to determine d R.
By No. 513, neglecting the squares and products of the inclinations of
the orbits, we have
R = ^-L cos. (v' — v) — ^ (£2 — 2 s / cos. (v' — v) + g'2)~ *
cos. (v" — v)
— 2 s f> cos. (v" — v)
If we develope this function in a series ordered according to the cosines
of v' — v, v" — v and their multiples ; we shall have an expression of
this form
COS. (V — V) -f- (*' (ft
cos. 2 (v' - v)
- (0)
(g, n (0) + ^"(ft f'O (1) cos. (v" - v) + p," (g, /') W cos. 2 (v" - v)
ft f") (3) COS. 3 (V7' — V) + &C. ;
whence we derive
^ I
cos. 2 (v' — v) + &c.
LCOS. 2 (v" — v) + &c.
, / A*' (ft f') (1) sin. (V —v) + 2 0 (f, £') W sin. 2(v' — v) + &c. 1
v \ + ^/(f , /') d)sin.(v"— v) + 2,u"(ft fx/> ^sin.2(v//_v) +&C. J .
Suppose, conformably to what observations indicate in the system of
the three first satellites of Jupiter, that n — 2 n' and n' — 2 n" are
very small fractions of n, and that their difference n — 2 n' — (n' — 2 n'y)
or n — 3 n' + 2 n" is incomparably smaller than each of them.
*
It results from the expressions of - , and of d v of No. 517, that the
action of// produces in the radius-vector and in the longitude of//, a very
sensible inequality depending on the argument 2 (n' t — n t + *' — e).
The terms relative to this inequality have the divisor 4 (n' — n) 2 — n 2,
BOOK I.] NEWTON'S PRINCIPIA. 175
or (n — 2 n') (3 n — 2 n'), and this divisor is very small, because of the
smallness of the factor n — 2 n'. We also perceive, by the consideration
of the same expressions, that the action of ^ produces in the radius-
vector, and in the longitude of //, an inequality depending on the argu
ment (n' t — n t + s' — E), and which having the divisor (n' — n) 2 — n' 2,
or n (n — 2 n'), is very sensible. We see, in like manner, that the action
of &" upon f! produces in the same quantities a considerable inequality
depending upon the argument 2 (n" t — n' t + *" — *')• Finally, we
perceive that the action of yJ produces in the radius-vector and in the
longitude of &" a considerable inequality depending upon the argument
n" t — n' t + t" — g. These inequalities were first recognised by obser
vations ; we shall develope them at length in the Theory of Jupiter's Sa
tellites. In the present question we may neglect them, relatively to other
inequalities. We shall suppose, therefore,
d g = [i! E' cos. 2 (n' t — n t + ?' — 2) ;
a v = 11! F' sin. 2 (n' t — n t + ••' — e) ;
If — ft" E" cos. 2(n" t — n't + s" — t')+ft G cos. (n t — nt + s'-— s)
a v' = (*" F" sin. 2(n"t — ri t + i' — £')+/"• H sin. (n' t — n t + «' — e)
d z" = p G' cos. (n" t — n' t + i" — *')
d v" = it," H' sin. (n" t — n' t + i" — e).
We must, however, substitute in the preceding expression of d R for
fj v> g> v/» *"» v//> the values of a 5 g, n t + s + 5 v, a' + d g', n' t+ «' + 5 V,
a" + 3 /', n" t + s// + 3 V", and retain only the terms which depend upon
the argument n t— 3 nr t + 2 n" t + £ — 3 t' + 2 s". But it is easy to see
that the substitution of the values of 8 ^ d v, 8 g", 3 v" cannot produce any
such term. This is not the case with the substitution of the values of
8 £ and 5 v' : the term (i! (g, g') W d v sin. (v7 — v) of the expression of
d R, produces the following,
,„
sin. (n t — 3 n' t + 2 n" t + t — 3 tf + 2 *").
This is the only expression of the kind which the expression of d R
<\
contains. The expressions of — , and of 3 v of No. 517, applied to the
action of //' upon &', give, retaining only the terms which have the divisor
n' — 2 n", and observing that n" is very nearly equal to ^ n',
(n' — 2 n") (3 n' — 2 n")
176 A COMMENTARY ON [SECT. XI-
__ 2E"
a' '
we therefore have
d n = »' »;n d * E-. | 2 (a' r° '" - (d • (t ? '") I
2 t • \ d a' / )
Xsin. (n t — 3n' t + 2n//t + s—3 g' + 2i//)= — I .^-?.
a 4
Substituting this value of — - in the values of — r-^ , -p-^- . • , • , and
a2 d t d t d t '
making for brevity's sake
we shall have, since n is very nearly equal to 2 n', and n' to 2 n",
^2 — 3.511 + 2.ilL = /3n'sin.(nt— 3n' t + 2n" t + e ~3 f
or more exactly
so that if we suppose
V = ^ — 3 ^ + 2 C' + • - 3 .' + 2 s",
we shall have
The mean distances n, a', a", varying but little as also the quantity n,
we may in this equation consider /3 n 2, as a constant quantity. Integrat
ing, we have
-M- ±dV
V c — 2 |8 n 2 cos. V
c being an arbitrary constant. The different values of which this con
stant is susceptible, give rise to the three following cases.
If c is positive and greater than + 2 (3 n 2, the angle V will increase
continually, and this ought to take place, if at the origin of the motion,
(n — 3 n' + 2 n") 2 is greater than + 2 /3 n 2 (1 + cos. V), the upper or
lower signs being taken according as (3 is positive or negative. It is easy
to assure ourselves of this, and we shall see particularly in the theory of
the satellites of Jupiter, that /3 is a positive quantity relatively to the three
first satellites. Supposing therefore + v = or — V, T being the semi cir
cumference, we shall have
d~
V c + 2 0 n 2 cos.
BOOK I.] NEWTON'S PRINCIPIA. 177
In the interval from -a — 0 to -a =r — , the radical V c + 2 /3 n 2 cos.
is greater than V 2 fi n 2, when c is equal or greater than 2 /3 n 2 ; we
have therefore in this interval -a > n t V 2 (3. Thus, the time t which the
i T
angle w employs in arriving from zero to a right angle is less than -- .
/w H 'r £ £>
The value of /3 depends upon the masses, w, /<*', /M," ; the inequalities ob
served in the three first satellites of Jupiter, and of which we spoke above,
give, between their masses and that of Jupiter, relations from whence it
results that -==— is under two years, as we shall see in the theory
of these satellites ; thus the angle •» would employ less than two years to
increase from zero to a right angle ; but the observations made upon Ju
piter's satellites, give since their discovery, -a constantly nothing or insen
sible; the case which we are examining is not therefore that of the three
first satellites of Jupiter.
If the constant c is less than + 2 /3 n 2, the angle V will not oscillate ;
it will never reach two right angles, if |8 is negative, because then the
radical V c — 2 j3 n z cos. V, becomes imaginary ; it will never be no
thing if J3 is positive. In the first case its value will be alternately greater
and less than zero ; in the second case it will be alternately greater and
less than two right angles. All observations of the three first satellites of
Jupiter, prove to us that this second case belongs to these stars ; thus the
value of /3 ought to be positive relatively to them ; and since the theory
of gravitation gives /3 positive, we may regard the phenomenon as a new
confirmation of that theory.
Let us resume the equation
dl = — -d " -
V c + 2 13 n 2 cos. w
The angle w being always very small, according to the observations,
we may suppose cos. -a = 1 — £ •&* • the preceding equation will give by
integration
tsr =: X sin. (n t V /3 + y)
X and y being two arbitrary constants which observation alone can deter
mine. Hitherto, it has not been recognised, a circumstance which proves
it to be very small.
From the preceding analysis result the following consequences. Since
the angle n t + 3 n' t + 2 n" t + s — 3 «' + It'' oscillates being some
times less and sometimes greater than two right angles, its mean value is
VOL. II. M
178 A COMMENTARY ON [SECT. XI.
equal to two right angles ; we shall therefore have, regarding only mean
quantities
n — 3 n' + 2 n" = 0
that is to say, that die mean motion of the Jlrst satellite, minus three times
that of the second, plus twice that of the third, is exactly and constantly
equal to zero. It is not necessary that this equality should subsist exactly
at the origin, which would not in the least be probable ; it is sufficient
that it did very nearly so, and that n — 3 n' + 2 n" has been less, ab
straction being made of the sign, than X n V j8 : and then that the mutual
attraction has rendered the equality rigorous.
We have next t — 3 s -f 2 i" equal to two right angles ; thus the mean
longitude of the first satellite, minus three times that of the second, pins twice
that of the third, is exactly and constantly equal to two right angles.
From this theorem, the preceding values of <§ /, and of 8 v' are reduci
ble to the two following
8 g = (p G — f" E") cos. (n t — n t + «' — 0
a v' = (i* H — P." F") sin. (n' t — n t + «' — «)•
The two inequalities of the motion of (i! due to the actions of fi and of
it*'', merge consequently into one, and constantly remain so.
It also results from this theorem, that the three first satellites can never
be eclipsed at the same time. They cannot be seen together from Jupi
ter neither in opposition nor in conjunction with the sun ; for the preced
ing theorems subsist equally relative to the synodic mean motions, and to
the synodic mean longitudes of the three satellites, as we may easily
satisfy ourselves. These two theorems subsist, notwithstanding the alter
ations which the mean motions of the satellites undergo, whether they
arise from a cause similar to that which alters the mean motion of the
moon, or whether from the resistance of a very rare medium. It is evi
dent that these several causes only require that there should be added to
the value of -* — r, a quantity of the form of -rrT , and which shall only
d t ' (it"
become sensible by integrations ; supposing therefore V = it — -a, and -a
very small, the differential equation in V will become
The period of the angle n t V jS being a very small number of years,
f\ 2 1
whilst the quantities contained in -p? are, either constant, or embrace
many ages; by integrating the above equation we shall have
BOOK I.] NEWTON'S PRINCIPIA. 179
62 4/
» = X sin. (n t V /3 + 7) — gn«dt«*
Thus the value of » will always be very small, and the secular equa
tions of the mean motions of the three first satellites will always be order
ed by the mutual action of these stars, so, that the secular equation of the
first, plus twice that of the third, may be equal to three times that of the
second.
The preceding theorems give between the six constants n, n', n",
s, e', t" two equations of condition which reduce these arbitraries to four ;
but the two arbitraries X and y of the value of or replace them. This
value is distributed among the three satellites, so, that calling p, p', p" the
coefficients of sin. (n t V /3 + 7) in the expressions of v, v', v", these
d 2 t d 2 T d 2 ?"
coefficients are as the preceding values of -7— if 5 JTY 5 ~A~I? '> an<* more
over we have p — 3 p' + 2 p" = X. Hence results, in the mean mo
tions of the three first satellites of Jupiter, an inequality which differs for
each only by its coefficients, and which forms in these motions a sort of
libration whose extent is arbitrary. Observations show it to be insen
sible.
53 1. Let us now consider the variations of the excentricities and of the
perihelions of the orbits. For this purpose, resume the expressions of
d f, d F, d f" found in 53 T : calling § the radius-vector of /* projected
upon the plane of x, y ; v the angle which this projection makes with the
axis of x ; and s the tangent of the latitude of <A above the same plane, we
shall have
x = P cos. v ; y = ° sin. v ; z = g s
whence it is easy to obtain
d RN /d Rx , 2X /d
x - z = ] + 8 > cos- v ~ s cos- v
d R
s S1U- v
d R
s sin- v -d7 - « s sm- v
/d Rx
- S COS. V ( -j-
\d v /
By 531, we also have
xdy — ydxrrcdt; xdz — zdxrrc'dt; ydz — zdy = c dt;
M2
ISO A COMMENTARY ON [SECT. XL
the differential equations in f, P, f " will thus become
df = -dy -*'<' +s')co,v ()_, i co.. v (
/dH\ I
+ s sm- v (ar) /
dRx)
L^
\d s / J \d s /
, „ ,. . d R
d y (1 + s2) sin.
d f / |Vdv g s
d R\ cos. v /d R\ s. sin. v /d
, .
. d t sm. v
The quantities c', c" depend, as we have seen in No. 531, upon the in
clination of the orbit of # to the fixed plane, in such a manner that they
become zero when the inclination = 0 ; moreover it is easy to see by the
nature of R that (—, — ) is of the order of the inclinations of the orbits ;
v. d s/
neglecting therefore the squares and products of these inclinations, the
preceding expressions of d f and of d f ', will become
, ,. i /d R\ j f • /d R\ . cos. v /d R
d f = ~- - d - - c d t ^sm- v
„. , /d R\ , / /d R\ sin. v /d R\ \
f = d x (av) + c d l lcos- v C-di) - - (ar, )S -
but we have
d x = d (g cos. v) ; d y = d (g sin. v); cdt=xdy — ydx = gsdv,
we therefore get
4 f = — [d s sin. v + 2 g d v cos. v} (^ — ) — f 2 d v sin. v (-T- ) j
d f = Jd g cos. v — 2 f d v sin. v} (-j— ) + ?s d v cos. v (-^ — ).
These equations are more exact, if we take for the fixed plane of x, y,
BOOK I.] NEWTON'S PRINC1PIA. 181
that of the orbit of p, at a given epoch ; for then c', c" and s are of the
order of the perturbing forces ; thus the quantities which we neglect, are
of the order of the squares of the perturbing forces, multiplied by the
square of the respective inclination of the two orbits of p and of /&'.
The values off, d f, d v, (-^ — V (-—, — \ remain clearly the same what
ever is the position of the point from which we reckon the longitudes ;
but in diminishing v by a right angle, sin. v becomes — cos. v, and cos. v
becomes sin. v ; the expression of d f changes consequently to that of
d f ' ; whence it follows that having developed, into a series of sines and
cosines of angles increasing proportionally with the times, the value of
d f, we shall have the value of d f ', by diminishing in the first the angles
i, i', *, »', 6 and 6' by a right angle.
The quantities f and f ' determine the position of the perihelion, and
the excentricity of the orbit ; in fact we learn from 531, that
f
tan. 1 = — r ;
I being the longitude of the perihelion referred to the fixed plane. When
this plane is that of the primitive orbit of ^, we have up to quantities of
the order of the squares of the perturbing forces multiplied by the square
of the respective inclinations of the orbits, I — •&, -a being the longitude of
the perihelion upon the orbit ; we shall therefore then have
P
tan. • .
which gives
cos. *r =r
V f°- + f/2 V f* + f'2
By 531, we then get
f / c> f c/>
/ .1*9 i L' / 9 i i.*// 9 (*lt *"* * V"
me = V i 2 + r 2 + i z , f " = :
c
thus c' and c" being in the preceding supposition of the order of the
perturbing forces, f" is of the same order, and neglecting the terms of the
square of these forces, we have
m e = V f« + f/2.
If we substitute for V f 2 + f *, its value m e, in the expressions of
sin. w, and of cos. w, we shall have
m e sin. « = f ; me cos. w = f ;
these two equations will determine the excentricity and the position of the
perihelion, and we thence easily obtain
m z. e d e = f d f + f ' d f ; m 2 e ' d « = f d f ' — f ' d f.
M3
182 A COMMENTARY ON [SECT. XI.
Taking for the plane of x, y that of the orbit of /A; we have for the
cases of the invariable ellipses,
- a (1 — e2) . _ g fc d v . e . sin, (v — «r) _
s " 1 + e cos. (v — tr) ' S ~' a(l — e2)
g 2 d v = a 2 n d t VI — e2 ;
and by No. 530, these equations also subsist in the case of the variable
ellipses ; the expressions of d f and of d f will thus become
d f = _ -==- £2 cos. v + | e cos. ™+\ e cos. (2 v — *)}
— - a ' n d t V 1 — e 2. sin. v .
d
df> = -- andt [2 sin. v+| e sin. <*+ £ e sin. (2 v — «r)J
VI — e2
+ a2 n d t V 1 — e 2. cos. v(-r— );
•
wherefore
andt . N c,. ,, X7 /d R\
e d * = -- 7T=1 sin' (v — «r) [2 + e cos. (v — *)} f-r- )
— 2
m V 1 — e
, / \
. (v — w\ ( -: — )
' \ d /
a2, n d t V 1 —e2 , /d R
- • cos.
m
e =
m V I — e
.\
m
This expression of d e may be put into a more commodious form in
some circumstances. For that purpose, we shall observe that
substituting for g and d § their preceding values, we shall have
but we have
P * d v = a 2 n d t V 1 — e2;
n d t [I + e cos. (v — *)}* m
d v = — s — >
(1-e2)'^
wherefore
,-i i • / x /d R
a2ndt V 1 — e~. sin. (v — -)•
e V 1 — e"
BOOK 1.] NEWTON'S PRINCIPIA. 183
the preceding expression of d e, will thus give
a n d t V 1— e2 /d Rx a (1 — e2)
p H p -— - . I — = - I — • - - — U IV.
m v/ m
We can arrive very simply at this formula, in the following manner
We have by No. 531,
d c /d Rx /d Rx /d l
but by the same No. c = V m a (1 — e2) which gives
d a V m a (1 -^ es) e d e V m a
d c = - ^ — s --- . , „ ;
2 a VI — ez
therefore
« da
m Vdv; '2a2
and then we have by No. 53 1
^ = - d R.
2 a2
We thus obtain for e d e the same expression as before.
535. We have seen in 532, that if we neglect the squares of the per
turbing forces, the variations of the principal axis and of the mean mo
tion contain only periodic quantities, depending on the configuration of
the bodies /*, /«/, ^", &c. This is not the case with respect to the varia
tions of the excentricities and inclinations : their differential expressions
contain terms independent of this configuration and which, if they were
rigorously constant, would produce by integration, terms proportional to
the time, which at length would render the orbits very excentric and
greatly inclined to one another ; thus the preceding approximations, found
ed upon the smallness of the excentricity and inclination of the orbits,
would become insufficient and even faulty. But the terms apparently
constant, which enter the differential expressions of the excentricities and
inclinations, are functions of the elements of the orbits ; so that they vary
with an extreme slowness, because of the changes they there introduce.
We conceive there ought to result in these elements, considerable inequa
lities independent of the mutual configuration of the bodies of the system,
and whose periods depend upon the ratios of the masses y.+ /a,', &c. to the
mass M. These inequalities are those which we have named secular in
equalities, and which have been considered in (520). To determine them
by this method we resume the value of d f of the preceding No.
d f =• ' {2 cos v + I e cos. * 4- A e cos. (2 v — *)} [—, — )
VI— e2 Wl v '
184 A COMMENTARY ON [SECT. XI.
— a 2 n d t V 1 — e2.sin
d
We shall neglect in the developement of this equation the square and
products of the excentricities and inclinations of the orbits ; and amongst
the terms depending upon the excentricities and inclinations, we shall re
tain those only which are constant : we shall then suppose, as in No. 515.
S = a(l + u,); / = a'(l + u/) ;
v = n t + £ -f v, ; v' = n7 t -f s + v/.
Again, if we substitute for R, its value found in 515; if we next con
sider that by the same No. we have,
d Rx a /d Rx /d
and lastly if we substitute for u/5 u/, v/} v/ their values — e cos. (n t+ 1 — «r),
— e' cos. (n' t + t' — «•'), 2 e sin. (n t + t — '•*), 2 e' sin. (nf t + «' — »')
given in No. 484, &c. by retaining only the constant terms of those which
depend upon the first power of the excentricities of the orbits, and ne
glecting the squares of the excentricities and inclinations, we shall find
that
— a ^ n d t. 5 j i A « + 1 a (^— ) } sin. Ji(n' 1— n t + e'— - s) + n t + *};
the integral sign belonging as in the value of R of 515, to all the whole
positive and negative values of i, including also the value of i = 0.
We shall have by the preceding No. the value of d f, by diminishing
in that of d f the angles i, «', *, •=/ by a right angle; whence we get
a (j! n d t ( /d A «»v . /d2 A <°>
-— . . e. cos. ~ a __ a* -
(
. e. cos. ~ ja
,.mi
-a/ndt e'. cosV A 0) + i
r /d A ^'\ i
+ a/Vndt. 2-j iA (i) + ^a {--; - J Vcos.£i (n-'t— n t+j'— s) + n t+s].
Let X, for the greater brevity, denote that part of d f, which is con
tained under the sign 2, and Y the corresponding part of d i'. Make also,
as in No. 522,
nn */n/
°' 1} = ~-r |a
BOOK I.] NEWTON'S PRINCIPIA. 18.5
then observe that the coefficient of e' d t sin. <JT', in the expression of d f,
is reducible to |0, Ij when we substitute for the partial differences in a',
their values in partial differences relative to a; finally suppose, as in 517,
that
e sin. zt — h ; e' sin. «' = h'
e cos. -or = 1 ; e' cos. «/ = \f
which gives by the preceding No. f =r m 1, f = m h or simply f = I,
¥ — hj by taking M for the mass-unit, and neglecting & with regard to
M ; we shall obtain
j£= (0, l).l-joTT.l'+aA*'nY;
< = - (0, 1). h + |0, 1|. h' - a yf n. X.
Hence, it is easy to conclude that if we name (Y) the sum of the terms
analogous to a /•*•' n Y, due to the motion of each of the bodies fj.', p", &c.
upon ^ ; that if we name in like manner (X) the sum of the terms analo
gous to — a fjt/ n X due to the same actions, and finally if we mark suc
cessively with one dash, two dashes, &c. what the quantities (X), (Y), h,
and 1 become relatively to the bodies fjf, A", &c. ; we shall have the fol
lowing differential equations,
dh
= 1(0,1) + (0,2) + &c.} 1 - [0,J 1' - JOTS) 1" - &c. + (Y);
~ = — J(0, 1) + (0, 2) + &c.J h + OH] h' + |OT2| h" + &c+ (X) ;
|£ = {(1, 0) + (1, 2) + &c.} 1' _ [T70| 1 - [172] 1" - &c. + (Y') •
~ - - J(l, 0) + (1, 2) + &c.} h' + O h + [iT2|h^+&c.+ (X/)
&c.
To integrate these equations, we shall observe that each of the quanti
ties h, 1, h', F, &c. consists of two parts ; the one depending upon the
mutual configuration of the bodies «•, //, &c. ; the other independent of
this configuration, and which contains the secular variations of these quan
tities. We shall obtain the first part by considering that if we regard
hat alone, h, 1, h', 1', &c. are of the order of the perturbing masses, and
consequently, (0, 1). h, (0, 1). 1, &c. are of the order of the squares of
186 A COMMENTARY ON [SECT. XL
these masses. By neglecting therefore quantities of this order, we shali
nave
d n _. /v\ . d * -. cv \ .
dT - (Y)' dT '
dh; __ m . dj[' _ t
d t " v '' d t "
wherefore,
h=/(Y)dt; l=/(X)dt; h'=/(Y')dt; &c.
If we take these integrals, not considering the variability of the ele
ments of the orbits and name Q what/(Y) d t becomes ; by calling 3 Q
the variation of Q due to that of the elements we shall have
/(Y)dt = Q-/5Q;
but Q being of the order of the perturbing masses, and the variations of
the elements of the orbits being of the same order, 5 Q is of the order of
the squares of the masses ; thus, neglecting quantities of this order, we
shall have
/(Y) d t = Q.
We may, therefore, take the integrals/ (Y) d t, / (X) d t, / (Y') d t,
&c. by supposing the elements of the orbits constant, and afterwards con
sider the elements variable in the integrals ; we shall after a very simple
method, obtain the periodic portions of the expressions of h, 1, h', &c.
To get those parts of the expressions which contain the secular inequa
lities, we observe that they are given by the integration of the preceding
differential equations deprived of their last terms, (Y), (X), &c. ; for it is
clear that the substitution of the periodic parts of h, 1, h', &c. will cause
these terms to disappear. But in taking away from the equations their
last terms, they will become the same as those of (A) of No. 522, which
we have already considered at great length.
536. We have observed in No. 532, that if the mean motions n t and
n' t of the two bodies & and X» are very nearly in the ratio of i' to i so
that V n' — in may be a very small quantity ; there may result in the
mean motions of these bodies very sensible inequalities. This relation of
the mean motions may also produce sensible variations in the excentrici-
ties of the orbits, and in the positions of their perihelions. To determine
them, we shall resume the equation found in 534,
an dt. VI — e2 /d R\ a (1 — e2) 7 _,
e d e = • . ( -r— ) S '- d R.
m \ d v / m
It results from what has been asserted in 515, that if we take for the
fixed plane, that of the orbit of /*, at a given epoch, which allows us to
BOOK I.] NEWTON'S PRINCIPIA. 187
neglect in R the inclination <p of the orbit of ^ to this plane; all the terms
of the expression of R depending upon the angle i' n' t — i n t, will be
comprised in the following form,
li! k cos. (i' n' t — i n t + i' «' — i t — g * — g/ J — g' ^),
i, i', g, g, g" being whole numbers and such that we have 0 = i'-i-g-g'-g".
The coefficient k has the factor e «. e' *' (tan. £ <p') *" ; g, g', g" being taken
positively in the exponents : moreover, if we suppose i and V positive, and
i' greater than i; we have seen in No. 515, that the terms of R which
depend upon the angle i' n' t — i n t are of the order i' — i, or of a su
perior order of two, of four, &c. units ; taking into account therefore only
terms of the order i' — i, k will be of the form e «. e' *' (tan. |- <ff) *". Q,
Q being a function independent of the excentricities and the inclination
of the orbits. The numbers g, g', g" comprehended under the symbol
cos., are then positive ; for if one of them, g for instance, be negative and
equal to — f, k will be of the order f + g' + g" ; but the equation 0 = i'
— i — g — g' — g" gives f + g' + g" = i' — i + 2 f ; thus k will be
of an order superior to i' — i, which is contrary to the supposition. Hence
J T> J T>
by No. 515, we have ( -, — ) = (—, — ) provided that in this last partial
J \dv/\d£/
difference, we make t — -a constant; the term of (-* — j corresponding
to the preceding term of R, is therefore
«/ (i + g) k sin. (i' n' t — i n t + i' «' — i e — g » — g J — g" 6').
The corresponding term of d R is
«,' 0 i n k d t sin. (i' n' t — i n t + V t' — i e — g =r — g' •*' — g" &'}.
Hence only regarding these terms and neglecting e 2 in comparison with
unity, the preceding expression of e d e, will give
ul a n d t Q k . . ,
d e = . 2 — sm. (i' n' t — i n t -f- i s — i t— g » — g' */ — g" r) ,
but we have
ge«-'. e'*'. (tan. |p')g//. Q= (^
integrating therefore we get
e = P7—. — . — . ( -.-— ^ cos. (i' n' t — 5 n t + i' t' — i i— g ^ — g' o-' — g" ^).
m (i'n — in) \d e/
The sum of all the terms of R, however, which depend on the angle
i' n' t — i n t. being represented by the following quantity
/<*'. P sin. (i' n' t — i n t + i' E' — i 0 + tt/ P' cos. (i' n' t — i n t + i' •'— i t)
the corresponding part of e will be
188 A COMMENTARY ON [SECT. XI.
This inequality may become very sensible, if the coefficient i' n' — i n
is very small, for it actually takes place in the theory of Jupiter and Sa
turn. In fact, it has for a divisor only the first power of i' n' — i n, whilst
the corresponding inequality of the mean motion, has for a divisor the se
cond power of this quantity, as we see in No. 532; butf— 1 — ) and f -, — )
being of an order inferior to P and P', the inequality of the excentricity
may be considerable, and even surpass that of the mean motion, if the
excentricities e and e' are very small ; this will be exemplified in the
theory of Jupiter's satellites.
Let us now determine this corresponding inequality of the motion of
the perihelion. For that purpose, resume the two equations
fdf+f'df fdf — f'df
ede = - -^- -, e'd^--^ -•
which we found in No. 534. These equations give
d f =r m d e cos. -a — m e d «. sin. ~;
thus with regard only to the angle
i' n' t — i n t + i' e' — is — g z, — g *' — g" 6',
we shall have
d f = (if. a n d t (^) cos. » sin. (i' n' t — i n t + i' «'— i «— g ~— g' »'— g"0
— m e d -a . sin. •&.
Representing by
— ii/. a n d t { (^) + k'} cos. (i' n' t— i n t + i' *'— i •— g ~— g' »'— g" 8),
the part of m e d •», which depends upon the same angle, we shall have
d f = (ii. a n d t { (^) + |k'} sin.(i' n' t-i n t + i' f'-i «— (g-l)*-gV-g"O
— ^"-^k'sin. (i'n't — int + iV — it— (g + l)»— g'*/ — g'Y).
It is easy to see by the last of the expressions of d f, given in the No.
534, that the coefficient of this last sine has the factor e e + l. e' g/ (tan. \ p)g" ;
k' is therefore of an order superior to that of (ir1-) by two units; thus,
(cl lc\
-j — J , we shall have
.andt /d k\ ... . , •/ / • „/ ' n" a\
cos. (i' n7 1 — i nt+iV — it — g» — g»- — g7 ff)
m \d e
for the term of e d w, which corresponds to the term
(jj k cos. i' n' t — \nt + \'i' — it — * —
BOOK I.] NEWTON'S PRINCIPIA. 189
of the expression of R. Hence it follows that the part of w, which cor
responds to the part of R expressed by
a,' P sin. (i' n' t — i n t + i' •' — i t} + (jJ P' cos. (Vn't — int+Vt'—i e),
is equal to
—r--r-r-^—\- • } ( i — ^ cos.(i'n't-int + i' s'-ii}- (—, — ^ sin.(i'n't-int + rY-ig) c >
m(i'n'-in)e t\de/ V d e /
we shall therefore, thus, after a very simple manner, find the variations
of the excentricity and of the perihelion, depending upon the angle
i' n' t — i n t + i' e' — i e. They are connected with the variation £ oi
the corresponding mean motion, in such a way that the variation of the
excentricity is
3in' Vde.dt
and the variation of the longitude of the perihelion is
i' n' — in /d A
Sine ' \d~e)'
The corresponding variation of the excentricity of the orbit of /«/, due
to the action of ^ will be
_!_ fj££-\
3i'n'. e'' Vde'.d J'
and the variation of the longitude of its perihelion, will be
i' n' — in /d
3 i' n' e' \<
and since by No. 532, £' = — ^ a, . £, the variations will be
fj> v a
i* V a. / d2 g \ , (i' n' — i n) /«, V a d £
3 i' . n'. y! V a' VdVTd t) ' a "srii^V'V^a'' ' dV '
When the quantity i7 n' — i n is very small, the inequality depending
upon the angle i' n' t — i n t, produces a sensible one in the expression
of the mean motion, amongst the terms depending on the squares of the
perturbing masses ; we have given the analysis of this in No. 532. This
same inequality produces in the expression of d e and of d =r, terms of
the order of the squares of the masses, and which, being only functions of
the elements of the orbits, have a sensible influence upon the secular
variations of these elements. Let us consider, in fact, the expression of
d e, depending on the angle i' n' t — in t.
By what precedes, we have
de = —
/*'. a n . d t
m
Ud P\ ,.. .
-T — ] cos. (r n t — i n t + i %' — is)
" ("d~e") ' Sin< (l/ »' t - i n t + i' «' -
190 A COMMENTARY ON [SECT. XI.
By No. 532 the mean motion n t, ought to be augmented by
-r ?' a " I 1 - \ Pcos. (i'n't— int+iV— is)— Fsin.(i' n' t-i n t+ i' i'— i «) I
(in— in)2.m I ;J
and the mean motion n' t, ought to be augmented by
3 fjf a n2. i /* V a fri .., . . . , ., . . .
— 7*7— F — !-n — • ' / / /-£P cos. (i n' t — i n t + i' t' — i 6) —
(i' n' — in) 2. m ^ V a'
F sin. (i' n' t — i n t + i' e' — i «)}.
In virtue of these augments, the value of d e will be augmented by the
function
3^a2. in3, dt ,. , . , , ., , 7 f „ /dP x „, /dP \ 1
i./^'V a' +i> v aM P. f-. — ) + P'f-i — ) J-;
I ^de/ \de/J
/ / /•/ / • \
a'. (I'n'-in)
and the value of d w will be augmented by the function
3 (if a 2. i n 3. d t . . / D /d PN , _, /d P\ )
5 — 8 7 //•/ / - S-TT^'fi^ Va'+ i> Va}. IP. (j— ) + Fl-T—J f .
3m1 v » (r nr — in)2, e I \d e / vde/J
In like manner we find that the value of d e' will be augmented by the
function
and that the value of d e' will be augmented by the function
d
These different terms are sensible in the theory of Jupiter and Saturn, and
in that of Jupiter's satellites. The variations of e, e', •&, *r' relative to the
angle i' n' t — i n t may also introduce some constant terms of the order of
the square of the perturbing masses in the differentials d e, d e', dw, and d*/,
and depending on the variations of e, e', w, •&' relative to the same angle.
This may easily be discussed by the preceding analysis. Finally it will
be easy, by our analysis, to determine the terms of the expressions of
e, », e', w which depending upon the angle i' n' t — i n t + \f *' — i e
have not i' n' — in for a divisor, and those which, depending on the same
angle and the double of this angle, are of the order of the square of the
perturbing forces. These different terms are sufficiently considerable in
the theory of Jupiter and Saturn, for us to notice them : we shall deve-
lope them to the extent they merit when we come to that theory.
537. Let us determine the variations of the nodes and inclinations of
the orbits, and for that purpose resume the equations of 53 1 ,
BOOK I.] NEWTON'S PRINCIPIA. 191
, R
dc =
, „
=
If we only notice the action of /i*7, the value of R of No. 513, gives
d R\ /d R
" x
RN /d R
Let however,
c'
the two variables p and q will determine, by No. 53 1, the tangent of the
inclination <p of the orbit of /*, and the longitude 6 of its node by means of
the equations
tan. <p = V p 2 -f q 2 ; tan. d = -_ .
Call p7, q', p", q", &c. what p and q become relatively to the bodies
/A7, At", &c. : we shall have by 531,
z = q y — p x ; z' = q' y — p7 x', &c.
The preceding value of p differentiated gives
d p J_ d c/7 — p d c
dt :: T' dt
substituting for d c, and d c77 their values we get
af = £ Kq — q') y / + (P' - P) x/ y? x
Hx'l + y'2 + z'1)1 J(X7_x)« + (y'_y)*-f(z<--z)
In like manner we find
= ^(P/ - P) x x7 + (q _ q7) x y7} X
192 A COMMENTARY ON SECT. XL
3
x'2 + y" +y*) KX' — x)*+(y'_ y)2+(z' — z)2}
If we substitute for x, y, x', y' their values g cos. v, o sin. v, %' cos. v',
g' sin. v', we shall have
(q— q') y y' + (p — p) *' y = q J3- s ?• icos- (v'+v) — cos. (v'— v)i
sn-
(p'— p) x x' + (q— q') x y' = j^- S $'• i™*- (v'+v) + cos. (v'— v)}
+ ^^^ • g ?• {sin. (v'+v) + sin. (v'— v)}.
Neglecting the excentricities and inclinations of the orbits, v, e have
s = a ; v = n t + f ; ? = a' ; v' = n' t + «' ;
which give
_ 1 __ ___ 1 _ _ 1_
(x/s+ y/f+ z/2)* Ux' — x)s + (y' — y)*+ (Z'_ Z)^f ~a/3
£a2 — 2 a a' cos. (n' t — n t + «' — s) + a/s] »
moreover by No. 516,
- ? - 5- = \ 2. B «. cos. i (n7 1— n t+*'— «)
{a2— 2 a a' cos. (n' t— n t + «'—*) +a' 2}^
the integral sign 2 belonging to all whole positive and negative values of
i, including the value i = 0 ; we shall thus have, neglecting terms of the
order of the squares and products of the excentricities and inclinations of
the orbits,
dp q m _
<& c a
. jsin. (n't+ nt+ «'+«)— sin. (n' t — nt + .'
m _» . {coSm (n/ 1 + n t + ,/ + g) _ cos. (n/ 1 _ n t+ ,/_,
cl t <& c a
c a
-q . /. a a'. 2. B « £cos.[(i+ 1) (n t— n t+i7— 0]
C
— cos.[(i+l) (n t— n t+e' — e) + 2nt+2*]}
=^ . /. a a'. 2. B W fsin.[(i+l) (iV t— n t+a'— i)J
C
— sin.[(i+l) (n't — n t+i'— 0+ 2nt+2«]|.
• * Jcos> (n/ 1 + n l + *' + £) + cos< (11/ 1-" n t+s/~£^
BOOK I.] NEWTON'S PRINCIPIA. 193
+ cl'— _9. ^ . [sin. (n' t + n t + s' +e) + sin. (n' t--nt + i/— i)}
•wl C U
+ p-p2-'. v!. a a'. 2. B W.{cos. [(i+ 1) (n' t— n t-H'— 0]
TP C
+ cos. [(i+1) (n't— n t+if—t) + 2
+ 2p3. ,(*'. a a'. 2. B W. £sin. [(i+1) (n't— n t+i'— «)]
T0 O
+ sin. [(i + 1) (n't— n t+/— s) + 2 n t+2 OJ-
The value i = — 1 gives in the expression of - — , the constant quan
tity -~ — - . /*'. a a' B (— !) : all the other terms of the expression of -~~
4 c d t
are periodic : denoting their sum by P, and observing that B (~ !) = B W
by 516, we shall have
i? = i.=L3. A*'. a a'. B<" + P.
at 4 c
By the same process we shall find, that if we denote by Q the sum of
all the periodic terms of the expression of-r-j" , we shall have
U L
.. .
d t 4 c
If we neglect the squares of the excentricities and inclinations of the
orbits, by 531, we have c r= V m a, and then supposing m = 1, we
have n 2 a 3 = 1 which gives c = — ; the quantity f/" a a" - thus be-
an 4 c
comes — - ^ - which by 526, is equal to (0, 1); hence we get
lH = (0, 1). (q'-q)+P;
^ = (0, 1). (p _ p') + Q.
Hence it follows that, if we denote by (P) and (Q) the sum of all the
functions P and Q relative to the action of the different bodies fjft p", &c.
upon A*; if in like manner we denote by (P), (Q'), (P"), (Q"), &c. what
(P) and (Q) become when we change successively the quantities relative
to p into those which are relative to /, /A", &c. and reciprocally ; we shall
have for determining the variables p, q, p', q', p", q", &c. the following
system of differential equations,
^Pt = — {(0, 1) + (0, 2) + &c.} q + (0, 1). q' + (U, 2) q"+ &c.+ (P) ;
VOL. II. N
194 A COMMENTARY ON [SECT. XI.
jS. ={(0, 1) + (0, 2) + &c.} p — (0, 1) p' - (0, 2) p" - &c. + (Q) ;
= — {(1, 0) + (1, 2) + &c.J q' + (1, 0) q + (1, 2) q"+ &c. + (F);
L'={(],0) + (1,2) + &c^p'-(l,0)p-(l,2)p"-&c.
&c.
The analysis of 535, gives for the periodic parts of p, q, p', q', &c.
p =/(P).dt; q =/(Q).dt;
p'=/(F).dt; q'=/(Q').dt;
&c.
We shall then have the secular parts of the same quantities, by inte
grating the preceding differential equations deprived of their last terms
(P), (Q), (P'), &c. ; and then we shall again hit upon the equations (C)
of No. 526, which have been sufficiently treated of already to render it un
necessary again to discuss them.
538. Let us resume the equations of No. 531,
V c2 + c"1 c"
tan. p = - ; tan. Q = —
c c
ivhence result these
c c"
— = tan. ® cos. 6 ; - = tan. 0 sin. i.
c c
Differentiating, we shall have
d tan. p = — {d c' cos. & + d c" sin. 6 — d c tan. <p]
C
d 6 tan. <p = - {d c" cos. 6 — d c' sin. 6}.
C
If we substitute in these equations for -y— , -y— , -r— , their values
/d Rx /d Rx /d Rx /d Rx /d Rx /d Rx ,e
V ( T — ) — x [ T — ) , z I T — ) — xl -, — ) , z ( -j— } — y ( i — ) , and for
J Vdx/ >-dy/' \dx/ vdz/' Vd y / 'Viz/'
these last quantities their values given in 534 ; if moreover we observe
that s = tan. <p sin. (v — 0), we shall have
d t tan. 0 cos. (v - 6) f /d Rx . , ., , /d R
_ __
. tan. p =
1 + s2dt . /d R
,. d t tan. ® sin. (v - 6) ( /d Rx . . . /d R
d 6 . tan. p = - -^- - |? . ( d - ) sm.(v-^)+ (^-
(1 + s2) dt . . .,/d Rx
— » - ! - L - sin. (v — tfH-T — J.
c » ' \ d s /
BOOK I.] NEWTON'S PRINCIPI A. 19 i
These two differential equations will determine directly the inclination
of the orbit and the motion of the nodes.
They give
gin. (v — 0} d tan. <p — d 6 cos. (v — 6} tan. <p = 0;
an equation which may be deduced from this
s = tan. <p sin. (v — 6} ;
in fact, this last equation being finite, we may (530) differentiate it whe
ther we consider <f> and d constant or variable ; so that its differential,
taken by only making <p and d vary, is nothing ; whence results the pre
ceding differential equation.
Suppose, however, that the fixed plane is inclined extremely little to the
orbit of /a, so that we may neglect the squares of s and tan. f>, we shall
have
, . t . . /d R\
d 6 tan. <p = --- sin. (v — 6} IT— );
c \ds J
by making therefore as before
p = tan. p sin. & ; q = tan. <p cos. 6 ;
we shall have, instead of the preceding differential equations, the follow
ing ones,
d t /d Rx
d q = --- cos. v . ( -T— ) ;
c \ d s /
d t . /d Rx
d p = -- sin. v . ( -j — ) ;
c \ d s /
But we have also
s = q sin. v — p cos. v
which gives
/dRx _ I /d Rx /d JRx _ \ /d Rx
\ds/ sin. v'vdq/' \ds/~ cos. v \dp/'
wherefore
d t d
d t/d
We have seen in 515 that the function R is independent of the po
sition of the fixed plane of x, y ; supposing, therefore, all the angles 01
that function referred to the orbit of //-, it is evident that R will be a
function of these angles and the respective inclination of two orbits, an
N2
196 A COMMENTARY ON [SECT. XI.
inclination we denote by p/. Let 6J be the longitude of the node of the
orbit of /jf upon the orbit of/*; and supposing that
ti! k (tan. p/) « cos. (i' n' t — i n t + A — g 6f)
is a term of R depending on the angle i' n' t — i n t, we shall have, by
527,
tan. p/ . sin. 6f = p' — p ; tan. p/ cos. 6/ = q' — q ;
whence we get
(tan. p/) « sin. g „/= iq'-q + (p'-P) V-H '-fr -q-(p'-p) V-
(tan. ,/) . COS. g «/= - - --q- (p'-p) V-H »
With respect to the preceding term of R, we shall have
(tip") = — S (tan< ?/) TV k • sin- H' n' t— in t + A— (g— 1) 0/J ;
=~g (tan* P'/)8~V k cos> **' n' ' ~ * n l + A— (S— ] ) '/!•
If we substitute these values in the preceding expressions of d p and
d q, and observe that very nearly c = — , we shall have
Substituting these values in the equation
s = q sin. v — p cos. v
we shall have
s= -- g * * * a n
m
* ; , * (tan. ?/)«-» sin. f i' n' t — i n t — v + A— (g -1) 6f}.
(in — i n) v
This expression of s is the variation of the latitude corresponding to
the preceding term of R : it is evident that it is the same whatever may
be the fixed plane to which we refer the motions of ^ and /V, provided that
it is but little inclined to the plane of the orbits ; we shall therefore thus
have that part of the expression of the latitude, which the smallness of the
divisor i' n' — in may make sensible. Indeed the inequality of the lati
tude, containing only the first power of this divisor, is in that degree
less sensible than the corresponding inequality of the mean longitude,
which contains the square of the same divisor ; but, on the other hand,
tan. <pf is then raised to a power less by one ; a remark analogous to that
which was made in No. 536, upon the corresponding inequality of the
excentricities of the orbits. We thus see that all these inequalities are
BOCK I.] NEWTON'S PRINCIPIA. 197
connected with one another, and with the corresponding part of R, by
very simple relations.
If we differentiate the preceding expressions of p and q, and if in the
values of -a-*- and ^ - we augment the angles n t and n' t by the inequa
lities of the mean motions, depending on the angle i' n' t — i n t, there
will result in these differentials, quantities which are functions only of the
elements of the orbits, and which may influence, in a sensible manner, the
secular variations of the inclinations and nodes although of the order of
the squares of the masses. This is analogous to what was advanced in
No. 536 upon the secular variations of the excentricities and aphelions.
539. It remains to consider the variation of the longitude t of the epoch.
By No. 531 we have
— d ^ { E n cos. ( v — w) + E «-' cos. 2 ( v — w) + &c.] ;
substituting for E p, E;V\ &c. their values in series ordered according to
O * J vJ
the powers of e, series which it is easy to form from the general expres
sion of E :'} (473) we shall have
d i = — 2 d e sin. (v — •*) + 2 e d w cos. (v — •&)
+ e d e \l + \ e2+&c.} sin. 2 (v— «•)— e2 d » {f + £ e2 + &c.}cos.2 (v— ~)
— e2 d e U +• &c.} sin. 3 (v — ») + e 3d «• U + &c.} cos. 3 (v — *r)
+ &c.
If we substitute for d e and e d •* their values given in 534, we shall
find, carrying the approximation to quantities of the order e * inclusively,
de = a2'nd Vl — eM2 — fecos. (v — „) + e 2cos. 2 (v —
in
a n d t . . . r, , x, /d R
. . . r, , x, /
. e . Sin. (V - nr) \ 1 + i e COS. (V - •at}\ [
^
m V 1 — e 2
The general expression of d t contains terms of the form
X k . n d t . cos. (i' n' t — i n t + A)
and consequently the expression of i contains terms of the form
T-. — ; - -. — sin. (i' n' t — i n t + A) ;
in — i n
but it is easy to be convinced that the coefficient k in these terms is of
the order i' — i, and that therefore these terms are of the same order as
those of the mean longitude, which depend upon the same angle. These
having the divisor (i' n' — in) *, we see that we may neglect the corre
sponding terms of f, when i' n' — i n is a very small quantity.
N3
198 A COMMENTARY ON [SECT. XL
If in the terms of the expression of d e, which are solely functions of the
elements of the orbits, we substitute for these elements the secular parts
of their values ; it is evident that there will result constant terms, and
others affected with the sines and cosines of angles, upon which depend
the secular variations of the excentricities and inclinations of the orbits.
The constant terms will produce, in the expression of E, terms propor
tional to the time, and which will merge into the mean motion p. As to
the terms affected with sines and cosines, they will acquire by integration,
in the expression of s, very small divisors of the same order as the per
turbing forces ; so that these terms being at the same time multiplied and
divided by the forces, may become sensible, although of the order of the
squares and products of the excentricities and inclinations. We shall see
in the theory of the planets, that these terms are there insensible; but in
the theory of the moon and of the satellites of Jupiter, they are very sen
sible, and upon them depend the secular equations.
We have seen in No. 532, that the mean motion of/,*, is expressed by
--//andt.rfR,
and that if we retain only the first power of the perturbing masses, d R
will contain none but periodic quantities. But if we consider the squares
arid products of the masses, this differential may contain terms which are
functions only of the elements of the orbits. Substituting for the elements
the secular parts of their values, there will thence result terms affected with
sines and cosines of angles depending upon the secular variations of the
orbits. These terms will acquire, by the double integration, in the ex
pression of the mean motion, small divisors, which will be of the order of
the squares and products of the perturbing masses; so that being both
multiplied and divided by the squares and products of the masses, they
become sensible, although of the order of the squares and products of the
excentricities and inclinations of the orbits. We shall see that these terms
are insensible in the theory of the planets.
540. The elements of p's orbit being determined by what precedes, by
substituting them in the expressions of the radius-vector, of the longitude
and latitude which we have given in 484, we shall get the values of these
three variables, by means of which astronomers determine the position of
the celestial bodies. Then reducing them into series of sines and cosines,
we shall have a series of inequalities, whence tables being formed, we may
easily calculate the position of ^ at any given instant.
This method, founded on the variation of the parameters, is very useful
BOOK I.] NEWTON'S PKINCIPIA. 199
in the research of inequalities, which, by the relations of the mean motions
of the bodies of the system, will acquire great divisors, and thence become
very sensible. This sort of inequality principally affects the elliptic ele
ments of the orbits ; determining, therefore, the variations which result
in these elements, and substituting them in the expression of elliptic mo
tion, we shall obtain, in the simplest manner, all the inequalities made
sensible by these divisors.
The preceding method is moreover useful in the theory of the comets.
We perceive these stars in but a very small part of their courses, and ob
servations only give that part of the ellipse which coincides with the arc
of the orbit described during their apparitions ; thus, in determining the
nature of the orbit considered a variable ellipse, we shall see the changes
undergone by this ellipse in the interval between two consecutive appari
tions of the same comet. We may therefore announce its return, and
when it reappears, compare theory with observation.
Having given the methods and formulas for determining, by successive
approximations, the motions of the centers of gravity of the celestial bo
dies, we have yet U) apply them to the different bodies of the solar system :
but the ellipticity of these bodies having a sensible influence upon the
motions of many of them, before we come to numerical applications, we
must treat of the figure of the celestial bodies, the consideration of which
is as interesting in itself as that of their motions.
SUPPLEMENT
TO
SECTIONS XII. AND XIII.
ON ATTRACTIONS AND THE FIGURE OF THE CELESTIAL BODIES.
541. The figure of the celestial bodies depends upon the law of gravi
tation at their surface, and the gravitation itself being the result of the at
tractions of all their parts, depends upon their figm'e; the law of gravi
ty at the surface of the celestial bodies, and their figure have, therefore, a
reciprocal connexion, which renders the knowledge of the one necessary
to the determination of the other. The research is thus very intricate^
N4
200 A COMMENTARY ON [SECT. XII. & XIII.
and seems to require a very particular sort of analysis. If the planets were
entirely solid, they might have any figure whatever ; but if, like the earth,
they are covered with a fluid, all the parts of this fluid ought to be dis
posed so as to be in equilibrium, and the figure of its exterior surface de
pends upon that of the fluid which covers it, and the forces which act
upon it. We shall suppose generally that the celestial bodies are covered
with a fluid, and on that hypothesis, which subsists in the case of the earth,
and which it seems natural to extend to the other bodies of the system of
the world, we shall determine their figure and the law of gravity at their
surface. The analysis which we propose to use is a singular application
of the Calculus of Partial Differences, which by simple differentiation, will
conduct us to very extensive results, and which with difficulty we should
obtain by the method of integrations.
THE ATTRACTIONS OF HOMOGENEOUS SPHEROIDS BOUNDED BY SURFACES
OF THE SECOND ORDER.
542. The different bodies of the solar system may be considered as
formed of shells very nearly spherical, of a density varying according to
any law whatever ; and we shall show that the action of a spherical shell
upon a body exterior to it, is the same as if its mass were collected at its
center. For that purpose we shall establish upon the attractions of sphe
roids, some general propositions which will be of great use hereafter.
Let x, y, z be the three coordinates of the point attracted which we
call ft ; let also d M be the element or molecule of the spheroid, and
x', y', z' the coordinates of this element; if we call o its density, § being a
function of x', y', z' independent of x, y, z, we shall have
d M = | . d x' . d y' . d z' .
The action of d M upon ft decomposed parallel to the axis of x and
directed towards their origin, will be
g d x' . d y' . d z' (x — xQ
KX — x')2 + (y — y')2 + (z — z')2}1
and consequently it will be equal to
s d x' . d y' . d z'
d . — -£ J ^--
(* — x')
dx
calling therefore V the integral
r- § d x' . d y' . d z'
V (x — x) 2 + (y - y ) 2 + (z — z') 2
extended to the entire mass of the spheroid, we shall have — [-. — J
BOOK L] NEWTON'S PRINCIPIA. 201
for the total action of the spheroid upon the point ,«, resolved parallel to
the axis of x and directed towards its origin.
V is the sum of the elements of the spheroid, divided by their respec
tive distances from the point attracted ; to get the attraction of the sphe
roid upon this point, parallel to any straight line, we must consider V as
a function of three rectangular coordinates, one of which is parallel to this
straight line, and differentiate this function relatively to this coordinate ;
the coefficient of this differential taken with a contrary sign, will be the
expression of the attraction of the spheroid, parallel to the given straight
line, and directed towards the origin of the coordinate which is parallel to
it.
i
If we represent by ft the function { (x — x') 2 + (y — y')2+ (z — z')2}"2 ;
we shall have
V = //3.f.dx'dy'dz'.
The integration being only relative to the variables x', y', z', it is evi
dent that we shall have
/d^Vx
(dO
But we have
0 =
^d x
v
in like manner we get
/d2 Vx /d2 Vx /d2 V
: \dx2/ \dy2/ Uz2
This remarkable equation will be of the greatest use in the theory of the fi
gure of the celestial bodies. We may present it under more commodious
forms in different circumstances ; conceive, for example, from the origin
of coordinates we draw to the point attracted a radius which we call g ;
let d be the angle which this radius makes with the axis of x, and w the
angle which the plane formed by § and this axis makes with the plane of
x, y; we shall have
x = P cos. 6 ; v = P sin. 6 cos. -a : z rr P sin. 0 sin. •& ;
* •/ a
whence we derive
z
s= Vx2 + y'+z2; cos.0= -7=f======; tan. * = -
itivt
;d 2
d y
thus we can obtain the partial differences of gt d, *-, relative to the varia-
r/d2Vx /d2VN
bles x, y, z, and thence get the values of ^j-^r) > \3~y*)
202 A COMMENTARY ON [SECT. XII. & XIII.
in partial differences of V relative to the variables P., 6, ». Since we shall
often use these transformations of partial differences, it is useful here to
lay down the principle of it. Considering V as a function of the variables
x, y, z, and then of the variables P, 0, «, we have
d PX /d Vv /d 0
._
-• +
To get the partial differences [r^Yi IT-IJ IT— j 5 we must make
vdx/ VI x/ \dx/
x alone vary in the preceding expressions of P, cos. 6, tan. w ; differentiat
ing therefore these expressions, we shall have
/d P\ /d &\ sin. 0 /d w\
( r~ ) = cos- * ; ( r ) = -- ; ( j - ) = ° ;
Vdx/ \dx/ g Vdx/
which gives
/d V\ ,/dV\ sin. tf /d V
( - ) = cos.
\dx/
Thus we therefore get the partial difference (-=- — j , in partial differ-
ences of the function V, taken relatively to the variables g, 6t •**. Differ
entiating again this value of f-j — J 5 we shall have the partial difference
j 2 "V7
( j — I )in partial differences of V taken relatively to the variables g, 0, w.
By the same process the values of (-r— F) an(^ ( . 2')may be found.
In this way we shall transform equation (A) into the following one:
>a«Vv
/d2Vv cos.*. /dVx Vdw'V ^•^'•gVy /m
°= \m + sinTT U J + inT + e rr' • * '
And if we make cos. 6 = m, this last equation will become
dm / 1 —
543. Suppose, however, that the spheroid is a spherical shell whose
origin of coordinates is at the center ; it is evident that V will only de
pend upon g, and contain neither m nor wt the equation (C) will therefore
give
whence by integration we get
BOOK I.] NEWTON'S PRINCIPIA. 203
A and B being two arbitrary constants. We therefore have
_(1_Y^ = i.
\ d £ / g2
— expresses, by what precedes, the action of the spherical shell upon
the point /«-, decomposed along the radius g and directed towards the
center of the shell ; but it is evident that the total action of the shell
'd_V;
dg
the total action of the spherical shell upon the point p.
First suppose this point placed within the shell. If it were at the center
itself, the action of the shell would be nothing ; we have therefore,
= 0, or — = 0,
d V
ought to be directed along this radius ; — (—. — ) expresses therefore
when § = 0, which gives B = 0, and consequently — -^ — ) = 0, what
ever £ may be ; whence it follows that a point placed in the interior of the
shell, suffers no action, or which comes to the same thing, it is equally at
tracted on all sides.
If the point //, is situated without the spherical shell, it is evident, sup
posing it infinitely distant from the center, that the action of the shell
upon the point will be the same, as if all the mass of the shell were con
densed at this center; calling, therefore Mthe mass of the shell, — (-,—
or —r will become in this case equal to — - , which gives B = M ; we have
S S
therefore generally relatively to exterior points,
/d Vx JV1
-(dg) '' ?
that is to say, the shell attracts them as if all its mass were collected at
its center.
A sphere being a spherical shell, the radius of whose interior surface ii
nothing, we see that its attraction, upon a point placed at or above its
surface, is the same as if its mass were collected at its center.
This result obtains for globes formed of concentric shells, varying in
density from the center to the circumference according to any law what
ever, for it is true for each of the shells : thus since the sun, the planets,
and satellites may be considered nearly as globes of this nature, they at
tract exterior bodies very nearly as if their masses were collected into
their centers of gravity. This is conformable with what has been found by
204 A COMMENTARY ON [SECT. XII. & XIII.
observations. Indeed the figure of the celestial bodies departs a lit-
lle from the sphere, but the difference is very little, and the error which
results from the preceding supposition is of the same order as" this sup
position relatively to points near the surface; and relatively to distant
points, the error is of the same order as the product of this difference by
the square of the ratio of the radii of the attracting bodies to their
distances from the points attracted; for we know that the considera
tion alone of the distance of the points attracted, renders the error of
the preceding supposition of the same order as tne square of this ratio.
The celestial bodies, therefore, attract one another very nearly as if their
masses were collected at their centers of gravity, not only because they
are very distant from one another relatively to their respective dimensions,
but also because their figures differ very little from the sphere.
The property of spheres, by the law of Nature, of attracting as if their
masses were condensed into their centers, is very remarkable, and we may
be curious to learn whether it also obtains in other laws of attraction.
For that purpose we shall observe, that if the law of gravity is such, that
a homogeneous sphere attracts a point placed without it as if all its mass
were collected at its center, the same result ought to obtain for a spherical
shell of a constant thickness; for if we take from a sphere a spherical
shell of a constant thickness, we form a new sphere of a smaller radius
with the remainder, but which, like the fonner, shall have the property of
attracting as if all its mass were collected at its center ; but it is evident,
that these two spheres can only have this common property, unless it also
belongs to the spherical shell which forms their difference. The problem,
therefore, is reduced to determine the laws of attraction according to which
a spherical shell, of an infinitely small and constant thickness, attracts an
exterior point as if all its mass were condensed into its center.
Let £ be the distance of the point attracted to the center of the spherical
shell, u the radius of the shell, and d u its thickness. Let d be the angle
wTiich the radius u makes with the straight line §, -a the angle which the
plane passing through the straight lines f, u, makes with a fixed plane
passing through £, the element of the spherical shell will be u 2 d u . d » .
d 6 sin. 0. If we then call f the distance of this element from the point at
tracted, we shall have
f2 = £2 — 2 g u cos. 6 + u*.
Represent by <p (f) the law of attraction to the distance f ; the action of
the shell's element upon the point attracted, decomposed parallel to g and
directed towards the center of the shell, will be
BOOK I.] NEWTON'S PRINCIPIA. 205
, , , . . f — u cos. 6 n
u 2 d u . d -a sin. 6 ~ 7= p (f ) ;
but we have
f — u cos. 6 _ /d f \
f ~ Vd g / '
which gives to the preceding quantity this form
(0;
wherefore if we denote fd f <f> (f) by <p, (f) we shall have the whole action
of the spherical shell upon the point attracted, by means of the integral
u2 d ufd -a d 6 sin. d. <p, (f ), differentiated relatively to f, and divided by
df.
This integral ought to be taken relatively to w, from » = 0 to v equal
to the circumference, and after this integration it becomes
2ffu2/d 0sin. 6 <p, (f ) ;
If we differentiate the value of f relatively to d, we shall have
fdf
d <J sin. 6 = ---- ;
S u
and consequently
' f. p, (f).
The integral relative to 0 ought to be taken from 6 = 0 to 6 = «r, and
at these two limits we have f = g — u, and f = § + u ; thus the integral
relative to f must be taken from f = g — utof=£ + u; let therefore
/f d f. p, (f) = ^ (f ), we shall have
2<!f.udu,, .,,. 2 T. u d u
, (f) =
The coefficient of d g, in the differential of the second member of this
equation, taken relatively to g, will give the attraction of the spherical
shell upon the point attracted ; and it is easy thence to conclude that in
nature where <f> (f ) = TT this attraction is equal to
4 it . u2 d u
~e~
That is to say, that it is the same as if all the mass of the spherical
shell were collected at its center. This furnishes a new demonstration of
the property already established of the attraction of spheres.
Let us determine <p (f ) on the condition that the attraction of the shell
is the same as if its mass were condensed into its center. This mass
is equal to 4 T. u8 d u, and if it were condensed into its center, its action
206 A COMMENTARY ON [SECT. XII. & XIII.
upon the point attracted would be 4 T. u * d u . <p (*) ; we shall therefore
have
d
integrating relatively to g, we shall get
•^ (g + u) — 4 ' (g — u) = 2 g u/d s . <p (g) + f U,
U being a function of u and Constants, added to the integral 2 ufd § p(g).
If we represent -^ (? + u) — -vj/ (g — u) by R, we shall have by differen
tiating the preceding equation
d ® M
But we have, by the nature of the function R,
dgR
du
wherefore
or
d.p(g) _ 1 /d 2 U
pf , .p(g _ 1 / x
g df 2u\du2/'
Thus the first member of this equation being independent of u and the
functions of g, each of its members must be equal to an arbitrary which we
shall designate by 3 A ; we therefore have
whence in integrating we derive
pg = Ag + -g-
B being a new arbitrary constant. All the laws of attraction in which a
sphere acts upon an exterior point placed at the distance g from its center,
as if all the mass were condensed into its center, are therefore comprised
in the general formula
it is easy to see in fact that this value satisfies equation (D) whatever may
be A and B.
If we suppose A = 0, we shall have the law of nature, and we see that
BOOK L] NEWTON'S PRINCIPIA. 207
in the infinity of laws which render attraction very small at great dis
tances, that of nature is the only one in which spheres have the properly
of acting as if their masses were condensed into their centers.
O
This law is also the only one in which a body placed within a spherical
shell, every where of an equal thickness, is equally attracted on all sides.
It results from the preceding analysis that the attraction of the spherical
shell, whose thickness is d u, upon a point placed in its interior, has the
expression
To make this function nothing, we must have
4 (u + f) — 4 (u — g) = g • U,
U being a function of u independent of g, and it is easy to see that this
T>
obtains in the law of nature, where <p (f ) = — 5 . But to show that it
takes place only in this law, we shall denote by «¥ (f) the difference of 4>
(f ) divided by d f, we shall also denote by-vj/' (f) the difference of -vj/ (f)
divided by d f, and so on ; thus we shall get, by differentiating twice suc
cessively, the preceding equation relatively to f,
-V' (u + g) — 4" (u — g) = o.
This equation obtaining whatever may be u and f, it thence results
that y (f ) ought to be equal to a constant whatever f may be, and that
therefore -ty" (f ) = 0. But, by what precedes,
•4/(f) = f.p;(f),
whence we get
4/"(f) = 8p(f) +fp'(f);
we therefore have
0 = 2p(f) + fp'(f);
which gives by integration
ic- \ B
MO = jr>
and consequently the law of nature.
554. Let us resume the equation (C) of No. 541. If this equation
could generally be integrated, we should have an expression of V, which
would contain two arbitrary functions, which we should determine by
finding the attraction of a spheroid, upon a point situated so as to facili
tate this research, and by comparing this attraction with its general ex
pression. But the integration of the equation (C) is possible only in some
particular cases, such as that where the attracting spheroid is a sphere,
which reduces this equation to ordinary differences; it is also possible in
208 A COMMENTARY ON [SECT. XII. & XIII.
the case where the attracting body is a cylinder whose base is an oval or
curve returning into itself, and whose length is infinite. This particular
case contains the theory of Saturn's ring.
Fix the origin of g upon the same axis of the cylinder, which we shall
suppose of an infinite length on each side of the origin. Naming g the
distance of the point attracted from the axis ; we shall have
S' = I "^ 1 — m 2«
It is evident that V only depends on §' and w, since it is the same for
all the points relatively to which these two variations are the same ; it
contains therefore only m inasmuch as g is a function of this variable.
This gives
/d V\ __ /d V\ /d P'\ %m /d V
\d m/ " ViTjp' vdrn/ " -v/r
'm2 /d2 Vx /dV
the equation (C) hence becomes
_ _ /
1— m2' \d'* " (1_mt)f ' VI / r
whence by integrating we get
V = <p{§ cos. •* + | V — 1 sin. «•} + %]/£/ cos. w — ^ V — 1 sin. »•} ;
<f (g') and >4/ (f7) being arbitrary functions of g', which we can determine
by seeking the attraction of the cylinder when » is nothing and when it
is a right angle.
If the base of the cylinder is a circle, V will be evidently a function of
f independent of v, the preceding equation of partial differences will
thus become
M
which gives by integrating,
d Vx H
H being a constant. To determine it, we shall suppose g' relatively to
the radius of the base of the cylinder extremely great, which supposition
permits us to consider the cylinder as an infinite straight line. Let A be
this base, and z the distance of any point whatever of the axis of the cy
linder, to the point where this axis is met by g' ; the action of the cylin
der considered as concentrated or condensed upon its axis, will be, paral
lei to g', equal to
/A f'. d z
— •
i »
BOOK I.] NEWTON'S PRINCIPIA. 209
the integral being taken from z = — oo to z = co ; this reduces the in
tegral to - - , ; which is the expression of — ( r~7") when g' is very con
siderable. Comparing this with the preceding one we have H = 2 A,
and we see that whatever is g', the action of the cylinder upon an exterior
. . 2 A
point, is —j- .
If the attracted point is within a circular cylindrical shell, of a constant
thickness, and infinite length, we shall have — ( , — "\ = — ; and since
\ a g / i
the attraction is nothing when the point attracted is upon the axis of the
shell, we have H =.- 0, and consequently, a point placed in the interior of
the shell is equally attracted on all sides.
545. We have thus determined the attraction of a sphere and of a
spherical shell : let us now consider the attraction of spheroids terminated
by surfaces of the second order.
Let x, y, z be the three rectangular coordinates of an element of the
spheroid ; designating d M this element, and taking for unity the density
of the spheroid which we shall suppose homogeneous, we shall have
dM = dx.dy.dz.
Let a, b, c be the rectangular coordinates of the point attracted by the
spheroid, and denote by A, B, C the attractions of the spheroid upon
this point resolved parallel to the axes of x, y, z and directed to the origin
of the coordinates.
It is easy to show that we have
A _ rrr (& x) d X . d y . d Z
{(a — x)2 + (b — y)2 + (c — z)2}^'
B =fff (b — y) dx. dy. dz
{(a — x)2 + (b — y)2 + (c — z)2}*'
C _. rrr (c — z) d x . d y . d z
( a— x)2 + (b — y)2 + (c — z)2}*'
All these triple integrals ought to be extended to the entire mass of the
spheroid. The integrations under this form present great difficulties,
which we can often in part remove by transforming the differentials into
others more convenient. This is the general principle of such trans
formations.
Let us consider the differential function Pdx.dy.dz, P being any
function whatever of x, y, z. We may suppose x a function of y and z
and of a new variable p : let p (y, z, p) denote this function ; in this case,
VOL. II. O
210 A COMMENTARY ON [SECT. XII. & XIII.
we shall have, making y and z constant, d x = /3 . d p, j3 being a function
of y, z and p. The preceding differential will thus become j8 . P . d p .
d y . d z ; and to integrate it, we must substitute «i P, for x, its value
0 (y, z, p).
In like manner we may suppose in this new differential, y = 0' (z, p, q),
q being a new variable, and 0' (z, p, q) being any function of the three
variables z, p and q. We shall have, considering z and p constant,
d y = /3' d q, /3' being a function of z, p, q ; the preceding differential
will thus take this new form /3 /3' P. d p . d q . d z, and to integrate it, we
must substitute in j3 P for y its value 0' (z, p, q).
Lastly we may suppose z equal to 0" (p, q, r), r being a new variable,
and 0" (p, q, r) being any function whatever of p, q, r. We shall have,
considering p and q constant, d z = {$" d r, ft" being a function of p, q, r ;
the preceding differential will thus become /3. /3'. j3". P . d p . d q . d r
and to integrate it, we must substitute in /3 . fi'. P for z its value 0" (p, q, r).
The proposed differential function is thence transformed to another rela
tive to the three new variables p, q, r, which are connected with the pre
ceding by the equations
x = 0 (y, z, p) ; y = <?' (z, p, q) ; z = 0" (p, q, r).
It only remains to derive from these equations the values of /3, /?, /3".
For that purpose we shall observe that they give x, y, z, in functions of
the variables p, q and r ; let us consider therefore the three first variables
as functions of the three last. Since $" is the coefficient of d r in the dif
ferential of z, taken by considering p and q constant, we have
*' = (ddf)-
|S' is the coefficient of d q, in the differential of y taken on the supposi
tion that p and z are constant ; we shall therefore have j6', by differen
tiating y on the supposition that p is constant, and by eliminating d r by
means of the differential of z taken on the supposition that p is constant,
and equating it to zero. Thus we shall have the two equations
d y = (TT:) d q + (!r?) d r
o =
d
d z
/u ZN j fa z\ j
(-7- ) d q -f- (-5— ) d r ;
\dq/ \d T/
which give
d y\ /d z\ /dy\ /d z
~ ~ ~
d y = d q X - •* - j
fPl
Vdr/
BOOK I.] NEWTON'S PRINCIPI A. 211
wherefore
dyx /dzv /dy
/yx /zv /y\
VdgJ Idr; " " VdrJ
Finally, /3 is the coefficient of d p, in the differential of x taken on the
supposition that y and z are constant. This gives the three following
equations
d x\ , dx\ , /d
If we make
\d p/ \d q/ Vd r/ \d p/ vdr
/dxx /d yx /d zx /d xx /d y\ /d z
^d Q/ Vdr/ xdp/ \d q/ xd p/ \d r
dp
dx
• ^-' \dpj vdq/~~VdT; vdq; \dp''
we shall have
« d p
d x = — s = C.
wliich gives
Vlx\ f^ fiz
Vdq; \dJ "" Mr
wherefore j8 . jS'. $" — s and the differential P. d x . d y . d z is transform
ed into E. P. dp. dq. dr; P being here what P becomes when we
substitute for x, y, z their values in p, q, r. The whole is therefore re
duced to finding the variables p, q, r such that the integrations may be
come possible.
Let us transform the coordinates x, y, z into the radius drawn from
the point attracted to the molecule, and into the angles which this ra
dius makes with given straight lines or with given planes. Let r be
this radius, p the angle which it forms with a straight line drawn through
the attracted point parallel to the axis of x, and let q be the angle which
o -2
212 A COMMENTARY ON [SECT. XII. & XIII.
its projection makes on the plane of y, z with the axis of y ; we shall
have
x = a — r cos. p ; y = b — r sin. p cos. q ; z = c — r sin. p sin. q.
We shall then find t = — r 2 sin. p, and the differential d x . d y . d z will
thus be transformed into — r 2 sin. p . d p . d q . d r : this is the expres
sion of the element d M, and since this expression ought to be positive
in considering sin. p, d p, d q, d r as positive, we must change its sign,
which amounts to changing that of £, and to making e = r 2 sin. p.
The expressions of A, B, C will thus become
A =fff<\ r d p d q . sin. p cos. p ;
B = fff& r dp d q . sin. 2 p cos. p ;
C = ffj 'd r dp d q. sin.2 p sin. q.
It is easy to arrive by another way at these expressions, by observing
that the element d M may be supposed equal to a rectangular parallele
piped, whose dimensions are d r, r d p and r d q sin. p, and by then observing
that the attraction of the element, parallel to the three axes of x, y, z is
d M d M dM
— g- cos. p ; — r-2- sin. p cos. q ; — — sin. p sin. q.
The triple integrals of the expressions of A, B, C must extend to the
entire mass of the spheroid : the integrations relative to r are easy, but
they are different according as the point attracted is within or without the
spheroid ; in the first case, the straight line which passing through the
point attracted, traverses the spheroid, is divided into two parts by this
point ; and if we call r and r7 these parts, we shall have
A =ff(r + r') d p d q. sin. p cos. p;
B = ff (r + r') d p d q . sin. 2 p cos. p ;
C = ff (r -f- r') d p d q . sin. 2 p sin. q ;
the integrals relative to p and q ought to be taken from p and q equal to
zero, to p and q equal to two right angles.
In the second case, if we call r, the radius at its entering the spheroid,
and r' the radius at its farther surface, we shall have
A =ff(v — r) d p d q . sin. p cos. p ;
B — ff(^ — r) d p d q . sin. 2 p cos. q ;
C = ff (r' — r) d p d q . sin. 2 p sin. q.
The limits of the integrals relative to p and to q, must be fixed at the
points where r' — r = 0, that is to say, where the radius r is a tangent
to the surface of the spheroid.
546. Let us apply these results to spheroids bounded by surfaces of the
BOOK I.] NEWTON'S PRINCIPIA. 213
second order. The general equation of these surfaces, referred to the
three orthogonal coordinates x, y, z is
OzrA + B.x + C.y + E.z+F. x2+H.xy + L.y2+M. xz+N. yz+O. z2.
The change of the origin of coordinates introduces three arbitraries,
since the position of this new origin relating to the first depends upon
three arbitrary coordinates. The changing the position of the coordi
nates around their origin introduces three arbitrary angles ; supposing,
therefore, the coordinates of the origin and position in the preceding
equation to change at the same time, we shall have a new equation of the
second degree whose coefficients will be functions of the preceding coeffi
cients and of the six arbitraries. If we then equate to zero the first
powers of the coordinates, and their products two and two, we shall de
termine these arbitraries, and the general equation of the surfaces of the
second order, will take this very simple form
x2 + m y2 + n zz = k2;
it is under this form that we shall discuss it.
In these researches we shall only consider solids terminated by finite
surfaces, which supposes m and n positive. In this case, the solid is an
ellipsoid whose three semi-axes are what the variables x, y, z become
k
when we suppose two of them equal to zero : we shall thus have k, — , — ,
V m
k
— for the three semi-axes respectively parallel to x, to y arid to z. The
1 1 3
solid content of the ellipsoid will be '
3 V m n
If, however, in the preceding equation we substitute for x, y, z their
values in p, q, r given by the preceding No., we shall have
r 2 (cos. 2 p + m sin.1 p cos.2 q + n sin. 2 p sin. * q)
— 2 r (a cos. p + m b sin. p cos. q + n c sin. p sin. q) = k 2-a 2-m bs-n cf ;
so that if we suppose
I = a cos. p + m b sin. p cos. q -f- n c sin. p sin. q;
L = cos. * p + m sin. 2 p cos. z q + n sin. 2 p sin. 2 q ;
R = I1 + (k2 — a2 — m b2 — n cs). L
we shall have
I + V R
"tr
whence we obtain r' by taking + , and r by taking — ; we shall there
fore have
21 , 2 V R
r + r'=T; r' - r = -j— .
O 3
214 A COMMENTARY ON [SECT. XII. & XIII.
Hence relatively to the interior points of the spheroid, we get
. si
L
A = 2 f f A P ' d q ' I ' sl"' P • cos- P -
TJ _ o r r d P • d q . I . sin. 2 p . cos, q
*fJ ~L~
, d p . d q . I . sin. 2 p . sin. q
~T~
and relatively to the exterior points
A g /• /» d p . d q . sin, p . cos, p V R
~~L~
p . d q . sin. 2 p cos, q V R
T '
r — 2 /*/* d p ' d q . sin. 2 p sin. q V R
yy ~~r~
the three last integrals being to be taken between the two limits which
correspond to R = 0.
547. The expressions relative to the interior points being the most
simple, we shall begin with them. First, we shall observe that the semi-
axis k of the spheroid does not enter the values of I and L ; the values of
A, B, C are consequently independent ; whence it follows that we may
augment at pleasure, the shells of the spheroid which are above the point
attracted, without changing the attraction of the spheroid upon this point,
provided the values of m and n are constant. Thence results the folloV-
ing theorem.
A point placed within an elliptic shell whose interior and exterior sur~
faccs are similar and similarly situated, is equally attracted on all sides.
This theorem is an extension of that which we have demonstrated in
542, relative to a spherical shell.
Let us resume the value of A. If we substitute for I and L their va
lues, it will become
A /. /.dp.dq.sin.p.cos.p.(acos.p + mbsin.pcos.q + ncsin.psin.q)
J J cos. 2 p + in sin. 2 p cos. 2 q + n sin. 2 p sin. 2 q
Since the integrals relative to p and q, must be taken from p and q
equal to zero, to p and q equal to two right angles, it is clear we have
generally f P d p . cos. p = 0, P being a rational function of sin. p and
of cos. z p ; because the value of p being taken at equal distances greater
and less than the right angle, the corresponding values of P . cos. p are
equal and have contrary signs ; thus we have
A = 2 a rr d p.dq.sin. p cos. z p ^
' J J cos. 2 p + m sin. 2 p cos 2 q -t- n sin " p sin. 2 q *
BOOK I.] NEWTON'S PRINCIPIA. 215
If we integrate relatively to q from q = 0 to q = two right angles, we
shall find
2 a *_ r d p . sin, p cos. * p
V m n / //_ 1 — m \ t , 1 — n \
/ L/ fi H cos' P 1 • (l + cos- P )
• -V \ m r/\ n */
an
integral which must be taken from cos. p rr 1 to cos. p = — 1. Let
cos. p = x, and call M the entire mass of the spheroid ; we shall have
-, 4«r.k3 , 4 cr 3M , ., .
by 545, M = — - and consequently -= = -j-r ; we shall there
's/ m n
fore have
3aM r
£\. •— — / — — T====
which must be taken from x = 0, to x = 1.
Integrating in the same manner the expressions of B, C we shall reduce
them to simple integrals ; but it is easier to get these integrals from the
preceding expression of A. For that purpose, we shall observe that this
expression may be considered as a function of a and of the squares k 2,
k ~ k 2
— , — of the semi-axes of the spheroid, parallel to the coordinates a, b, c
m n
of the point attracted ; calling therefore k' 2 the square of the semi-axis
parallel to b, and consequently k' 2. m, and k' 2 n the squares of the two
other semi-axes, B will be a similar function of b, k'*, k' 2 m, k'2 — ; thus
to get B we must change in the expression of A, a into b, k into k' or
k .1 n . . ,
—. — , m into — . and n into — , which gives
v/ m m m
m^. x2dx
Let
t
x =
m + (1 — m). t
we shall have
„ 3bM r t2! dt
M r _ t2! dt _
y- / j 7 5
/ (i+i^. ,«)"(! +!^l£.tf
N m / \ n /
an integral relative to t which must be taken, like the integral relative to x
O4-
216 A COMMENTARY ON [SECT. XII. & XIII.
from t = 0 to t = 1, because x = 0 gives t = 0 and x = 1, gives t = 1
Hence it follows that if we suppose
+ X2x2). (1 + x'2x2)'
we shall have
„ __ 3 b M
~~~~
If we change in this expression, b into c, X into X' and reciprocally, we
shall have the value of C. The attractions A, B, C of the spheroid, par
allel to its three axes are thus given by the following formulas
_ 3aM ,-, w _ 3 b M /d.xF\ r _ 3 c M
7" ^5 ( ~a~~
We may observe that these expressions obtaining for all the interior
points, and consequently for those infinitely near to the surface, they also
hold good for the points of the surface.
The determination of the attractions of a spheroid thus depends only
on the value of F ; but although this value is only a definite integral, it
has, however, all the difficulty of indefinite integrals when X and X' are
indeterminate, for if we represent this definite integral, taken from x = 0
to x = 1, by <p (Xs, X' z), it is easy to see that the indefinite integral will
be x 3 <p (X x 2, X' 2 x z), so that the first being given, the second is likewise
given. The indefinite integral is only possible in itself when one of the
quantities X, X' is nothing, or when they are equal : in these two cases,
the spheroid is an ellipsoid of revolution, and k will be its semi-axis of
revolution if X and X' are equal. In this last case we have
^ /• x z d x 1 .
==/i+x'x' = r^x-tan'"^-
To get the partial differences f—V— — ), (— ^ — ; — J, which enter the
expressions of B, C, we shall observe that
but when X = X', we have
/d . x F\ _ /d . x' Fx d_x _ d_x-
V d X ) * \ d x' / ; x >/ '
wherefore
Substituting for F its value, we shall have
d . X
BOOK I.] NEWTON'S PRINCIPIA. 217
we shall therefore have relatively to ellipsoids of revolution, whose semi-
axis of revolution is k,
A 3a.M., :.,
A = j-^ — 3- (X — tan. -1 X) ;
3 b.M/
1-4 — I tart — ' > —
~2k3. xAia
C = 3 c' M
548. Now let us consider the attraction of spheroids upon an exterior
point. This research presents greater difficulties than the preceding be
cause of the radical V R which enters the differential expressions, and
which under this form renders the integrations impossible. We may ren
der them possible by a suitable transformation of the variables of which
they are functions ; but instead of that method, let us use the following
one, founded solely upon the differentiation of functions.
If we designate by V the sum of all the elements of the spheroid divided
by their respective distances from the point attracted, and x, y, z the co
ordinates of the element d M of the spheroid, and a, b, c those of the
point attracted, we shall have
V = f JM
J V (a — x)2 + (b — y)2 + (c — z)2'
Then designating, as above, by A, B, C the attractions of the spheroid
parallel to the axes of x, y, z, and directed towards their origin, we shall
have
A=/ (a — x). dM
{(a. — x) 2 + (b — y) 2 + (c —
In like manner we get
'd V,
whence it follows that if we know V, it will be easy thence to obtain by
differentiation alone, the attraction of a spheroid parallel to any straight
line whatever, by considering this straight line as one of the rectangular
coordinates of the point attracted ; a remark we have already made in
541.
The preceding value of V, reduced into a series, becomes
fi i 2 a x+ 2 b y+ 2 c z — x 2 — y !
J TVT |*+-¥« „ 2 _L U2 _L „*
v=/
" '+&C.
This series is ascending relatively to the dimensions of the spheroid.
218 A COMMENTARY ON [SECT. XII.&X11I.
and descending relatively to the coordinates of the point attracted. If we
only retain the first term, which is sufficient when the attracted point is
at a very great distance, we shall have
Y- M
V a2 + b2 + c2'
M being the entire mass of the spheroid. This expression will be still
more exact, if we place the origin of coordinates at the center of gravity
of the sphere ; for by the property of this center we have
/ x. d M = 0 ; / y. d M = 0 ; / z. d M = 0 ;
so that if we consider a very small quantity of the first order, the ratio
of the dimensions of the spheroid to its distance from the point attracted,
the equation
V a2 + b2 + c2
will be exact to quantities nearly of the third order.
We shall now investigate a rigorous expression of V relatively to ellip
tic spheroids.
549. If we adopt the denominations of 544, we shall have
V =/ =fSS* d r d p d q sin. p = £//(r'2 — r 2) d p d q. sin. p,
Substituting for r and r' their values found in 544, we shall have
v - rr d p . d q sin, p. I . V R
: 2JJ ' L2
Let us resume the values of A B, C relative to the exterior points, and
given in 546,
* /» d p . d q sin, p cos, p V R
B = 2 /yyd p . d q sin. g p cos, q V R.
0 = 2/7" d P • d q sin- 2 P sin- q v R
Since at the limits of the integrals, we have V R = 0, it is easy to see
that by taking the first differences of.V, A, B, C relatively to any of the
six quantities a, b, c, k, m, n, we may dispense with regarding the varia
tions of the limits ; so that we have, for example,
for the integral
/d p sin. p I V R
LT-
BOOK I.] NEWTON'S PRINCIPIA. 210
is towards these limits, very nearly proportional to R 2, which renders
equal to zero, its differential at these limits. Hence it is easy to see by
differentiation that if for brevity we make
aA + bB + cC = F;
we shall have between the four quantities B, C, F, and V the following
equation of partial differences,
We may eliminate from this equation, the quantities B, C, F by means
of their values
d Vx /d
We shall thus get an equation of partial differences in V alone. Let
therefore
4*r.k3 .._
V = - == .v = M . v,
3 V m n
M being by 545, the mass of the elliptic spheroid ; and for the variables
m and n let us here introduce 6 and &• which shall be such that we have
1 — m i o 1 — ni2
6= - .k2; 9= - .k2;
m n
6 will be the difference of the square of the axis of the spheroid parallel
to y and the square of the axis parallel to x ; -a will be the difference of
the square of the axis of z and the square of the axis of x ; so that if we
take for the axis of x, the smallest of the three axes of the spheroid, V 0
and V -a will be its two excentricities. Thus we shall have
V being considered in the first members of those equations as a function
of a, b, c, k, m, n ; and v being considered in their second members as a
function of a, b, c, 6, «> k.
220 A COMMENTARY ON [SECT. XII. & XIII.
If we make
n fd v\ , i /d v\ . /d v\
Q = a(dli)+b(db)+c(d-c)i
1 T^
we shall have F = — M Q, and we shall get the values of k(-r , V
chansing in the Preceding values of k
-j — J , v into — Q. Moreover V and F are homogeneous functions in
a, b, c, k, V d, V -a of the second dimension, for V being the sum of the
elements of the spheroid, divided by their distances from the point at
tracted, and each element being of three dimensions, V is necessarily of
two dimensions, as also F which has the same number of dimensions as
V ; v and Q are therefore homogeneous functions of the same quantities
and of the dimension — 1 ; thus we shall have by the nature of homo
geneous functions,
an equation which may be put under this form
We shall have in like manner
then, if in equation (1) we substitute for V, F and their partial differences;
k2 k2
if moreover we substitute .• , . — . for m and ,— 9 — - for n, we shall have
k2 + d k2 + w
550. Conceive the function v expanded into a series ascending rela
tively to the dimensions k, V 6, V -a of the spheroid, and consequently
descending relatively to the quantities a, b, c : this series will be of the
following form :
v = U <°> + U (1' + U ® + U ^ + &c. ;
U (0), U (1), &c. being homogeneous functions of a, b, c, k, V 6, V &, and
separately homogeneous relatively to the three first and to the three last
BOOK I.]
NEWTON'S PRINCIPIA.
221
of these six quantities; the dimensions relative to the three first always
decreasing, and the dimensions relative to the three last increasing con
tinually. These functions being of the same dimension as v, are all of the
dimension — 1.
If we substitute in equation (2) for v its preceding expanded value ; if
we call s the dimension of U (i) in k, V t), V •&, and consequently — s — 1
its dimension in a, b, c ; if in like manner we name s' the dimension of
|J(' + i) in k? V 6, V **, and consequently — s' — 1 its dimension in a, b,
c ; if we then consider that by the nature of homogeneous functions we
have
we shall have, by rejecting the terms of a dimension superior in k, V 0,
V ar to that of the terms which we retain,
U a + 1) = _k
(3)
s'.
This equation gives the value of U (i + 1\ by means of U (i) and of its
partial differences ; but we have
(a2 + b2 + C2)2
since, retaining only the first term of the series, we have found in 548, that
v = M ..
(a2 + b2 + c2) 2
Substituting therefore this value of U (0) in the preceding formula, we
shall get that of U (1> ; by means of that of U (1) we shall have that of U (2)
and so on. But it is remarkable that none of these quantities contains k:
for it is evident by the formula (3) that U(0), not containing U(n, does
not contain it ; that U (1) not containing it, U (2) will not contain it, and so
on ; so that the entire series U (0) + U (1) + &c. is independent of k, or
which is the same thing ( = 0. The values of v, —
222 A COMMENTARY ON [SECT. XII. & XIII.
— (cTc) ' aie t^erefore the same f°r all elliptic spheroids similarly si
tuated, and which have the same excentricities V 0, V & ; but — M H-^
^d a/
AT /d v\ ,. . /d v\
- M Vd~j~) » — M ^ j-^J > express by 548,the attractions of the spheroid
parallel to its three axes; therefore the attractions of different elliptic
spheroids which have the same center, the same position of the axes and
the same excentricities, upon an exterior point, are to one another as their
masses.
It is easy to see by formula (3) that the dimensions of U <°>, U C1>, U C2>,
&c. in V6 and V •*, increase two units at a time, so that s = 2 i, s' = 2 i
moreover we have by the nature of homogeneous functions
this formula will therefore become
By means of this equation, we shall have the value of v in a series very
convergent, whenever the excentricities V d, V •& are very small, or when
the distance Va2 + bz + c2of the point attracted from the center of
the spheroid is very great relatively to the dimensions of the spheroid.
If the spheroid is a sphere, we shall have 0 = 0, and « = 0, which
give U (1) = 0, U (2) = 0, &c. ; wherefore
V = U W = 1 ;
V a2 + b2 + ca
and
M
V =
V a2 + b2 + c2
whence it follows that the value of V is the same as if all the mass of the
sphere were condensed into its center, and that thus, a sphere attracts any
exterior point, as if all its whole mass were condensed into its center ; a
result already obtained in 542.
551. The property of the function of v being independent of k, fur
nishes the means of reducing its value to the most simple form of which it
is susceptible ; for since we can make k vary at pleasure without changing
this value, provided the spheroid retain the same excentricities, V 6 and
BOOK I.] NEWTON'S PEIINCIPIA. 223
V *r, we may suppose k such that the spheroid shall be infinitely flatten
ed, or so contrived that its surface pass through the point attracted. In
these two cases, the research of the attractions of the spheroid is rendered
more simple; but since we have already determined the attractions of elliptic
spheroids, upon points at the surface, we shall now suppose k such that
the surface of the spheroid passes through the point of attraction.
If we call k', m', n' relatively to this new spheroid what in 545, we
named k, m, n relatively to the spheroid we there considered ; the condi
tion that the point attracted is at the surface, and that also a, b, c are the
coordinates of a point of the surface, will give
a* + m'bz + n' c2 = k2;
and since we suppose the excentricities V 6 and V w to remain the same,
we shall have
whence we obtain
k'2 , k'2
YYV ^— • tl """ — — — —•— — — "
HI — 1/0 I »> ll 1/0 , '
we shall therefore have to determine k', the equation
It is easy hence to conclude that there is only one spheroid whose sur
face passes through the point attracted, 6 and -a remaining the same. For
if we suppose, which we always may do, that 6 and •» are positive, it is
clear that augmenting in the preceding equation, k' 2 by any quantity which
we may consider an aliquot part of k/2, each of the terms of the first
member of this equation, will increase in a less ratio than k' 2 ; therefore
if in the first state of k' 2, there subsist an equality between the two mem
bers of this equation, this equality will no longer obtain in the second
state ; whence it follows that k' 2 is only susceptible of one real and posi
tive value.
Let M' be the mass of the new spheroid, and A', B7, C' its attractions
parallel to the axes of a, b, c ; if we make
1 _ m' 1 — n'
_ _ \ 2 . . — ^' *
m' n
' ~ J V(l + x2. x2). (1 + x". x*);
by 547, we shall have
_ 3 a M' F B, = 3b JV
221 A COMMENTARY ON [SECT. XII. & Xlll.
Changing in these values of A', B', C', M' into M, we shall have by
the preceding No., the values of A, B, C relatively to the first spheroid
but the equations
1 — m' 1 — n'
m' n'
give
,/2
• k —
> 5 _ * - / 2 __ W
= p-8; -k/2;
k' 2 being given by equation (5) which we may put under this form
we shall therefore have
3 a M «*/»- 3b M/d.xF^ 3_cM 1 " ~
— 1/3 — • Jc J -D — - i . „ I r— J 5 ^ — i / j
These values obtain relatively to all points exterior to the spheroid, and
to extend them to those of the surface, and even to the interior points
we have only to change k' to k.
If the spheroid is one of revolution, so that 6 = w, the formula (5)
will give
2 k/2 = a2+b2 + c2 — 6 + V(a2 + b2+c2 — 6) 2 + 4 a2.
and by 547. we shall have
3 a M /, a
~ k'~\Ts( *
3 b M >.
„
r
3 c M
Thus we have terminated the complete theory of the attractions of el
liptic spheroids ; for all that remains to be done is the integration of the
differential expression of F, and this integration in the general sense is
impossible, not only by known methods, but also in itself. The value of F
cannot be expressed in finite terms by algebraic, logarithmic or circular
quantities ; or which it tantamount, by any algebraic function of quantities
whose exponents are constant, nothing or variable. Functions of this kind
being the only ones which can be expressed independently of the symbol
J] all the integrals which cannot be reduced to such functions, are impos
sible in finite terms.
If the elliptic spheroid is not homogeneous, and if it is composed of
elliptic shells varying in position, excenlricity and density according to
any law whatever, we shall have the attraction of one of its shells, by de-
BOOK I.] NEWTON'S PRINCIPIA. 225
termining as above the difference of the attractions of two homogeneous
elliptic spheroids, having the same density as the shell, one of which shall
have for its surface the exterior surface of the shell, and the other the in
terior surface of the shell. Then summing this differential attraction, we
shall have the attraction of the whole spheroid.
THE DEVELOPEMENT INTO SERIES, OF THE ATTRACTIONS OF ANY
SPHEROIDS WHATEVER.
552. Let us consider generally the attractions of any spheroids what
ever. We have seen in No. 547, that the expression V of the sum of the
elements of the spheroid, divided by their distances from the attracted
points, possesses the advantage of giving by its differentiation, the attrac
tion of this spheroid parallel to any straight line whatever. We shall see
moreover, when treating of the figure of the planets, that the attraction of
their elements presents itself under this form in the equation of their equi
librium ; thus we proceed particularly to investigate V.
Let us resume the equation of No. 548,
v - r dM
J V (a — x) 2 + (b — y) 2 + (c — z) «'
a, b, c being the coordinates of the point attracted; x, y, z those of the
element d M of the spheroid ; the origin of coordinates being in the in
terior of the spheroid. This integral must be taken relatively to the va
riables x, y, z, and its limits are independent of a, b} c; hence we shall
find by differentiation,
an equation already obtained in 541,
Let us transform the coordinates to others more commodious. For
that purpose, let r be the distance of the point attracted from the origin
of coordinates ; 0 the angle which the radius r makes with the axis of a ;
•<* the angle which the plane formed by the radius and this axis, makes
with the plane of the axis of a, and of b ; we shall have
a = r cos. 6 ; b = r sin. 6 cos. 6 ; c = r sin. 6 sin. -a.
If in like manner we name R, tf, -a' what r, d, -a become relatively to
the element d M of the spheroid ; we shall have
x = R cos. ff ; y = R sin. & cos. -a' ; z = R sin. 6'. sin. -a'.
Moreover, the element d M of the spheroid is equal to a rectangular
parallelepiped whose dimensions are d R, R d 0', R d -a' sin. 6', and con-
VOL. II. P
226 A COMMENTARY ON [SECT. XII. & XIII.
sequently it is equal to g. R2. d R. d tf. d »'. sin. 0', g being its density; we
shall thus have
V - fff g R *. d R . d (f. d »' sin. tf_ _ __
JJJ ^ ra — g r R £cos> (?. cos. 0'+sin. 0 sin. f cos. (t/ — »)J + R2'
the integral relative to R must be taken from R = 0 to the value of R at
the surface of the spheroid ; the integral relative to */ must be taken from
•a' = 0 to •a' equal to the circumference ; and the integral relative to V
must be taken from 6' = 0 to (f equal to the semi-circumference. Differ
entiating this expression of V, we shall find
0 - f!\ , cos^ ,d Vx \d»-* /d 2. r \
~ \d t*) + sin. '&' \d~JJ ""• sin.2<J + r ( d r2
an equation which is only equation (1) in another form.
If we make cos. 6 = m, we may give it this form
' * ( '
We have already arrived at these several equations in 541.
553. First, let us suppose the point attracted to be exterior to the sphe
roid. If we wish to expand V into a series, it ought in this case, to de
scend relatively to powers of r, arid consequently to be of this form
u«*>
Substituting this value of V in equation (3) of the preceding No., the
comparison of the same powers of r will give, whatever i may be
It is evident from the integral expression alone of V that U (i) is a ra
tional and entire function of m, V 1 — m 2. sin. -or, and V 1 — m~2. cos. w,
depending upon the nature of the spheroid. When i = 0, this function
becomes a constant ; and in the case of i = 1, it assumes the form
H m + H' V 1 — m2. sin. *• + H" V 1 — m 2. cos. w ;
H, H', Hff being constants.
To determine generally U « call T the radical
^^^ __ 1 _
Vr* — 2 R r ic
BOOK I.] NEWTON'S PRINCIP1A. 227
we shall have
dm
This equation will still subsist if we change 6 into 6', * into •*', and re
ciprocally ; because T is a similar function of 01, •& and of 0, -a.
If we expand T, in a series descending relatively to r, we shall have
TJ
Q W being, whatever i may be, subject to the condition that
0 = _
dm / — m
and moreover it is evident, that Q (i) is a rational and entire function of m,
and V 1 — m2 . cos. (•&' — ») : Q (i) being known, we shall have U (l) by
means of the equation
U « =fg R(i + 2) . d R . d »' . d ff . sin. 6' . Q «.
Now suppose the point attracted in the interior of the spheroid : we
must then develope the integral expression of V, in a series ascending re
latively to r, which gives for V a series of the form
V = v (0> + r . v (1> + r 2 . v (2) + r 3 . v C3> + &c.
v (l) being a rational and whole function of m, V I — m 2 . sin. » and
VI — m z cos. «-, which satisfies the same equation of partial differences
that U (i) does ; so that we have
dm / 1 — m
To determine v (i), we shall expand the radical T into a series ascending
according to r, and we shall have
O W r r ~
T = ^ + Q «. ^2 + Q(2)- ^3 + &c-
the quantities Q (0), Q U), Q (2), &c. being the same as above ; we shall
therefore get
/•g.d R.dw'. dO'.sin. ' «
~
~
But since the preceding expression of T is only convergent so long as
R is equal to or greater than r, the preceding value of V only relates to the
shells of the spheroid, which envelope the point attracted. This point
being exterior, relatively to the other shells, we shall determine that part
of V which is relative to them by the first series of V.
P2
228 A COMMENTARY ON [SECT. XII. & XIII.
554. First let us consider those spheroids which differ but very little
from the sphere, and determine the functions U (0-, U (1), U (2), Sac. v (°),
v(1>, v(2>, &c. relatively to these spheroids. There exists a differential
equation in V, which holds good at their surface, and which is remarkable
because it gives the means of determining those functions without any in
tegration.
Let us suppose generally, that gravity is proportional to a power n of
the distance ; let d M be an element of the spheroid, and f its distance
from the point attracted; call V the integraiyf n + 1 d M, which shall ex
tend to the entire mass of the spheroid. In nature we have n = — 2,
/•d M
it becomes J — p — , and we have expressed it in like manner by V in the
preceding Nos. The function V possesses the advantage of giving, by its
differentiation, the attraction of the spheroid, parallel to any straight line
whatever ; lor considering f as a function of the three coordinates of the
point attracted perpendicular to one another, and one of which is parallel
to this straight line. Call r this coordinate, the attraction of the spheroid
1 f
along r and directed towards its origin, will bey. f n . f-f — }• d M. Con
sequently it will be equal to (-, — j , which, in the case of nature,
becomes — ( — ) , conformably with what has been already shown.
Suppose, however, that the spheroid differs very little from a sphere of
the radius a, whose center is upon the radius r perpendicular to the sur
face of the spheroid, the origin of the radius being supposed to be arbi
trary, but very near to the center of gravity of the spheroid; suppose,
moreover, that the sphere touches the spheroid, and that the point at
tracted is at the point of contact of the two surfaces. The spheroid is
equal to the sphere plus the excess of the spheroid above the sphere ; but
we may conceive this excess as being formed of an infinite number of
molecules spread over the surface of the sphere, these molecules being
supposed negative wherever the sphere exceeds the spheroid; we shall
therefore have the value of V by determining this value, 1st, relatively to
the sphere ; 2dly, relatively to the different molecules.
Relatively to the sphere, V is a function of a, which we denote by A ;
if we name d m one of the molecules of the excess of the spheroid above
the sphere, and f its distance from the point attracted ; the value of V rela-
BOOK L] NEWTON'S PRINCIPIA. 229
tive to this excess will be/. f n + l . d m ; we shall therefore have, for the
entire value of V, relative to the spheroid,
V = A+/. fn + i.dm.
Conceive that the point attracted is elevated by an infinitely small
quantity d r, above the surface of the spheroid and the sphere upon r or a
produced ; the value of V, relative to this new position of the attracted
point, will become
A will increase by a quantity proportional to d r, and which we shall re
present by A' . d r. Moreover, if we name 7 the angle formed by the two
radii drawn from the center of the sphere to the point attracted, and to
the molecule d m, the distance f of this element or molecule from the point
attracted, will be in the first position of the point, equal to
V 2 a2 (1 —cos. 7) ;
in the second position it will be
V (a + d r) 2 — 2 a (a -f- d r) cos. 7 + a 2,
or
the integral/, f n + 1 d m, will thus become
{' + ^-
we shall therefore have
substituting for/, f n + ! . d m, its value V — A. we shall have
f (n + 1} A n+ 1
In the case of nature, the equation (1) becomes
The value of V relative to the- sphere of radius a, is, by 550, equal to
-~- , which gives A = ^a ; A' = — ^~ ; we shall therefore
get
*'
We must here observe that this equation obtains, whatever may be the
position of the straight line r, and even in the case where it is not perpen-
r 3
230 A COMMENTARY ON [SECT. XII. & XIII.
dicular to the surface of the spheroid, provided that it passes very near its
center of gravity, for it is easy to see that the attraction of the spheroid,
resolved parallel to these straight lines, and which, as we have seen, is
equal to — (~TT) > *s> whatever may be their position, always the same, to
quantities nearly of the order of the square of the excentricity of the
spheroid.
555- Let us resume the general expression of V of 553, relative to a
point attracted exterior to the spheroid,
U&» , U<« . U<8> ,
V = — + -77- + "73 - + &c.
the function U (i) being, whatever i may be, subject to the equation of par
tial differences
dm / 1 — m2
By differentiating the value of V relatively to r, we have
/d Vx TJ(°> , 2U(1> , 3 U® .
— (-i— ) = — T M § 4 r- &c.
v d r / r 2 r 3 r 4
Let us represent by a (1 + ay) the radius drawn from the origin of
r to the surface of the spheroid, « being a very small constant coefficient,
whose square and higher powers we shall neglect, and y being a function
of m and » depending on the nature of the spheroid. We shall have to
41 IT 3.
quantities nearly of the order «, V = — ; whence it follows that in the
A 3
preceding expression of V, 1st, the quantity U (0) is equal to — — plus a very
small quantity of the order «, and which we shall -denote by U' W) ;
2dly, that the quantities U Cl), U (2), &c. are small quantities of the order a.
Substituting a (1 + a y) for r in the preceding expressions of V and of
>— f-r — V and neglecting quantities of the order a2, we shall have rela
tively to an attracted point placed at the surface
i T-r /* a / • \ > ^^ i ^^ ' r ^^ • O
If we substitute these values in equation (2) of the preceding No. we
shall have
2 , - U/(0) 3 ' U U) 5 U (*] + I*?. ^^ &C
fft'^~~a a2 a3 a4
BOOK I.] NEWTON'S PRINCIPIA. 231
It thence follows that the function y is of this form
y = Y<°> + YW + Y<8> + &c.
the quantities Y (0), Y W, Y (2\ &c. as well as U (0>, U (1>, &c. being subject
to the equation of partial differences
m
this expression of y is not therefore arbitrary, but it is derived from the
developement of the attractions of spheroids. We shall see in the follow
ing No. that y cannot be thus developed except in one manner only ; we
shall therefore have generally, by comparing similar functions,
(i) _ 4 av . + 3 Y p) .
' 2 i + 1
whence, whatever r may be, we derive
To get V, therefore, it remains only to reduce y to the form above de
scribed ; for which object we shall give, in what follows, a very simple
method.
If we had y = Y (i), the part of V relative to the excess of the spheroid
above the sphere whose radius is a, or which is the same thing, relative to
a spherical shell whose radius is a, and thickness a a y, would be
-TO~-~I — j\ 'i + i 5 this value would consequently be proportional to y,
and it is evident that it is only in this case that the proportionality can
subsist.
556. We may simplify the expression Y(0) + Y(1> + Y® + &c. of y,
and cause to disappear the two first terms, by taking for a, the radius of a
sphere equal in solidity to the spheroid, and by fixing the arbitrary origin
of r at the center of gravity of the spheroid. To show this, we shall ob
serve that the mass M of the spheroid supposed homogeneous, and of a
density represented by unity, is by 552, equal to/R2 d R d m d w, or to
^./R'3 d m d », R' being the radius R produced to the surface of the
spheroid. Substituting for R' its value a (1 + a y) we shall have
M = i-3 aa3dmd*r.
All that remains to be done, therefore, is to substitute for y its value
Y(0) + Y(1) + &c. and then to make the integrations. For this purpose
here is a general theorem, highly useful also in this analysis.
r 1
232 A COMMENTARY ON [SECT. XII. & XIII.
" If Y (i) and Z w be rational and entire functions of m, V 1 — m 2 . sin. -a
" and V 1 — m2. cos. *r, which satisfy the following equations :
0= _ —
dm
" we shall have generally
«/Y(1). Z«'>.dmd*r = 0,~
" whilst i and i' are whole positive numbers differing from one another.
" the integrals being taken from m = — 1 to m = 1, and from » = 0
" to » = 2 »."
To demonstrate this theorem, we shall observe that in virtue of the first
of the two preceding equations of partial differences, we have
/Y ».
a
m . d
_m
But integrating by parts relatively to m we have
and it is clear that if we take the integral from m = — 1 to m = 1, the
second member of this equation will be reduced to its last term. In like
manner, integrating by parts relatively to w, we get
and this second member also reduces to its last term, when the integral
BOOK I.] NEWTON'S PRINCIPIA. 233
/d Y (i\
is taken from w =r 0 to «r = 2 *, because the values of Y (l), (— 1 - },
N Cl -at /
Z(% (— , — \ are the same at these two limits; thus we shall have
/Y«. Z^.dm. d » =
dm
whence we derive, in virtue of the second of the two preceding equations
of partial differences,
/ Y «. Z O1). d m . d w=1Tjjl±l)- ./ Y W. Z M. d m . d * ,
we therefore have
0 =/Y«. Z« dm. d *,
when i is different from i'.
. Hence it is easy to conclude that y can be developed into a series of
the form Y(0> + Y(1> + Y® + &c. in one way only; for we have
generally
fy . Z « d m d « = / Y «. Z « d m . d ™ ;
If we could develope y into another series of the same form, Y/0) +
Y/ U) + Y7 ® + &c. we should have
/y.Z»> =/¥,«. Z«dm.d^;
wherefore
/Y, W. Z «). d m d tr rr /Y W. Z «) d m . d tr.
But it is easy to perceive that if we take for Z (l) the most general
function of its kind, the preceding equation can only subsist in the case
wherein Y, (i> = Y (i) ; the function y can therefore be developed thus in
only one manner.
If in the integraiy y d m . d w, we substitute for y its value Y (0) + Y (1)
-f Y® + &c., we shall have generally 0 — f Y (i) d m . d », i being
equal to or greater than unity ; for the unity which multiplies d m . d «
is comprised in the form Z ^D', which extends to every constant and quan
tity independent of m and *. The integraiy y d m . d •* reduces there
fore toy Y(0) d m . d w, and consequently to 4 T Y(0) ; we have there
fore
M = f era3 + 4 air a3. Y ^ ;
thus, by taking for a, the radius of the sphere equal^in solidity to the sphe
roid, we shall have Y(0) = 0, and the term Y(0) will disappear from the
expression of y.
234 A COMMENTARY ON [SECT. XII. & XIII.
The distance of the element d M, or R 2 . d R d m . d w, from the
plane of the meridian from whence we measure the angle w, is equal to
R V 1 — m2 . sin. *; the distance of the center of gravity of the sphe
roid from this plane, will be therefore/ R 3 d R d m . d »• VI — m 2. sin. *r,
and integrating relatively to R, it will be ^/R' 4 d m . d -a VI — m2 sin. •*,
R' being the radius R produced to the surface of the spheroid. In like
manner the distance of the element d M from the plane of the meridian
perpendicular to the preceding, being R V 1 — m 2 . cos. *, the distance
of the center of gravity of the spheroid from this plane will be \ f R/4
clm.dw. V I — m2. cos. *. Finally, the distance of the element d M
from the plane of the equator being m, the distance of the center of gra
vity of the spheroid from this plane will be \f R' 4 m . d m . d «. These
functions m, V I — m 2 . sin. », V 1 — m 2 . cos. w, are of the form Z(I>,
Z (1) being subject to the equation of partial differences
J ]
+ 2Z
dm / \ — m '
If we conceive R' 4 developed into the series N (0) + N (1) + N ® + &c.
N (i) being a rational and entire function of m, VI — m"2 . sin. ?r,
V 1 — m 2 . cos. "vr, subject to the equation of partial differences.
d
dm y 1 — m
the distances of the center of gravity of the spheroid, from the three
preceding planes, will be, in virtue of the general theorem above demon
strated,
i/N<«. dm. d». V 1 — m2. sin. *r,
4/N (1) . d m . d *> . V 1 — m2 . cos. * ;
. d m. d ».
N C1) is, by No. 553, of the form A m + B VI — m2 . sin. -a -f
C V 1 — m 2 . cos. w, A, B, C being constants ; the preceding distances
will thus become -^ . B, -^ . C, -^- . A. The position of the center of
o o o
gravity of the spheroid, thus depends only on the function N C1). This
gives a very simple way of determining it. If the origin of the radius R'
is at the center; this origin being upon the three preceding planes, the
distances of the center of gravity from these planes will be nothing. This
gives A = 0, B = 0, C = 0; therefore N (1) = 0.
BOOK I.] NEWTON'S PRINCIPIA 235
These results obtain whatever may be the spheroid : when it is very
little different from a sphere, we have R' = a (1 + « y), and R'4 =
a4 (1 + 4 a y) ; thus, y being equal to Y(0) + Y(1> + Y® + &c., we
have N C1) — 4 a a 4 Y (l\ the function Y (1' disappears, therefore, from the
expression of y, when we fix the origin of R' at the center of gravity of
the spheroid.
557. Now let the point attracted be in the interior of the spheroid, we
shall have by 553
V = v <°> + r . v (1> + r 2 . v & + r 3 v (3> -f &c.
r d R . d J . d (f . sin. tf . Q ^
v uj — . j i_l .
Suppose that this value of V is relative to ashell whose interior surface is
spherical and of the radius a, and the radius of whose exterior surface is
a (1 — «y); the thickness of the shell is a a y. If we denote by y' what
y becomes when we change Q, -a into <)', &', we may, neglecting quantities
of the order a 2, change r into a, and d R into a a y', in the integral ex
pression of v W ; thus we shall have
v W = -j^/y d w' . d (f . sin. (f . Q «.
a
Relatively to a point placed without the spheroid, we have, by 553,
v - u(0) H!l!
~~r~ + T~ ~*~ C'}
U(i) =fRl + 2. dR.d~r.dff. sin. 8. Q (i>.
If we suppose this value of V relative to a shell, whose interior and ex
terior radii are respectively a, a (1 + a y), we shall have
U (i1 = «. a' + './y. d */. d 6f. sin. V. Q «>;
wherefore
U-W
y W —
We have by 555
U w - -
therefore
i a w A .1
2i + 1
4 a 9 Y W
(2i+ iFaT
which gives
( r r 2
I +3~a' "^"Sa"2'
To this value of V we must add that which is relative to the spherical
shell of the thickness a — r which envelopes the attracted point plus that
which is relative to the sphere of radius r, and which is below the same
236 A COMMENTARY ON [SECT. XII. & XIII.
point If we make cos. tf = m', we shall have, relatively to the first of
the two parts of V,
r d R . d */ . d m' . Q w
v *'' ~ / - _ •
-J Ri-.l
an integral which, relative to m', must be taken from m' = — 1 to m == 1
Integrating relative to R, from R = r to R = a, we shall have
m. ;
But we have generally, by the theorem of the preceding No.,
yd •&' . d m'. Q (i) = 0 when i is equal to or greater than unity; when
i = 0, we have, by 553, Q (n = 1 ; moreover the integration relative to
«/ must be taken from -of = 0 to •&' = 2 <K ; we shall therefore have
v<°> = 2 * (a2 — r2).
This value of v (0) is that part of V which is relative to the spherical shell
whose thickness is a — r.
The part of V which is relative to the sphere whose radius is r is equal
to the mass of this sphere, divided by the distance of the attracted point from
4 .. 2
its center : it is consequently equal to — - — . Collecting the different
9
parts of V,we shall have its whole value
. (4)
Suppose the point attracted, placed within a shell very nearly spherical,
whose interior radius is
a + a a fY«» + Y«> + Y® + &c.}
and whose exterior radius is
a' + « a' [Y'W + Y'W + Y'® + &c.}
The quantities a a Y {0) and a af Y' (0) may be comprised in the quanti
ties a, of. Moreover, by fixing the origin of coordinates at the center of
gravity of the spheroid whose radius is
a+ « a fY<°> + Y«> + &c.$,
we may cause Y (I) to disappear from the expression of this radius ; and
then the interior radius of the shell will be of this form,
a + aa {Y® + Y® + &c.},
and the exterior radius will be of the form,
a' + « a' £Y/(1> -f Y'® + &c.}.
We shall have the value of V relative to this shell, by taking the differ
ence of the values of V relative to two spheroids, the smaller of which
shall have for the radius of its surface the first quantity, and the greater
BOOK I.] NEWTON'S PR1NCIPIA. 237
the second quantity for the radius of its surface ; calling therefore A . V,
what V becomes relatively to this shell, we shall have
If we wish that the point placed in the interior of the shell, should be
equally attracted on all sides, A . V must be reduced to a constant inde
pendent of r, 6, zr ; for we have seen that the partial differences of A . V,
taken relatively to these variables, express the partial attractions of the
shell upon the point attracted ; we therefore, in this case have Y' (1) = 0,
and generally
Y' W = f-£-V-2. y 0).
>• a /
so that the radius of the interior surface being given, that of the exterior
surface will be found.
When the interior surface is elliptic, we have Y (3) = 0, Y (4) = 0, &c.
and consequently Y/(3) = 0, Y/(4) = 0; the radii of the two surfaces, in
terior and exterior, are therefore
aU + «Y<*}; a'{l + «Y»>J;
thus we see that these two surfaces are similar and similarly situated,
which agrees with what we found in 547.
558. The formulas (3), (4) of Nos. 555, and 557, comprehend all the
theory of the attractions of homogeneous spheroids, differing but little from
the sphere; whence it is easy to obtain that of heterogeneous spheroids,
whatever may be the law of the variation of the figure and density of their
shells. For that purpose let a ( 1 + a y) be the radius of one of the shells
of a heterogeneous spheroid, and suppose y to be of this form
Y<°> + Y'1' + Y<2> -f- &c.
the coefficients which enter the quantities Y (0), Y (1), &c. being functions
of a, and consequently variable from one shell to another. If we differ
entiate relatively to a, the value of V given by the form (3) of No. 555 ;
and call g the density of the shell whose radius is a (1 + « y), § being a
function of a only ; the value of V corresponding to this shell will be, for
an exterior attracted point,
this value will be, therefore, relatively to the whole spheroid,
.; . . (5)
the integrals being taken from a = 0 to that value of a which subsists at
the surface of the spheroid, and which we denote by a.
238 A COMMENTARY ON [SECT. XII. & XliL
To get the part of V relative to an attracted point in the interiorwf the
spheroid, we shall determine first the part of this value relative to all the
shells to which this point is exterior. This first part is given by formula
(5) by taking the integral from, a = 0 to a = a, a being relative to the
shell in which is the point attracted. We shall find the second part of V
relative to all the shells in the interior of which is placed the point attract
ed, by differentiating the formula (4) of the preceding No. relatively to a;
then multiplying this differential by £, and taking the integral from a = a,
to a = a, the sum of the two parts of V will be its entire value relative to
an interior point, which sum will be
~ Y«> + &c.. (G)
the two first integrals being taken from a = 0 to a = a, and the two last
being taken from a = a to a = a; after the integrations, moreover, we
must substitute a for r in the terms multiplied by «, and — - —*• for
— in the term - — f P d . a 3.
r 3 rj
559. Now let us consider any spheroids whatever. The research of
their attraction is reduced, by 553, to forming the quantities U (i) and v ^ ,
by that No. we have
U« r=/gRi + 2. d Rdm'dt*'. Q«;
in which the integrals must be taken from R = 0 to its value at the sur
face, from m' = — 1 to m' = 1, and from •&' — 0 to •*/ = 2 it.
To determine this integral, Q W must be known. This quantity may
be developed into a finite function of cosines of the angle •& — «/, and of
its multiples. Let /3 cos. n («r — »') be the term of Q W depending on
cos. n (a — «/), £ being a function m, m'. If we substitute for Q (i) its
value in the equation of partial differences in Q (i) of No. 553, we shall
have, by comparing the terms multiplied by cos. n (& — »'), this equation
of ordinary differences,
R w
Q (i) being the coefficient of - . + t , in the developement of the radical
1
V r " — 2 Rr\m m'+ V 1 — m' 2 . V~l — m 2 . cos. (* — *'} + R !
BOOK I.J NEWTON'S PRINCIPIA. 239
The term depending on cos. n (&• — */), in the developement of this
radical, can only result from the powers of cos. (& — &'), equal to n, n-f-2,
n + 4, &c. ; thus cos. (a — «/) having the factor V I — m 2, /3 must have
the factor (1 — m 2) ^. It is easy to see, by the consideration of the de
velopement of the radical, that (3 is of this form
— m. . m
If we substitute this value in the differential equation in J39 the compari
son of like powers of m will give
A«'l (i-n-2s+2).(i-n-2S + 1) s_
2 s (2 i — 2 s + 1)
whence we derive, by successively putting s = 1, s = 2, &c. the values of
A (l), A (2), and consequently,
/
(
__
n-n--n--n--n--n-
2.4.6(2i — l)(2i — 3)(2i — 5)
A is a function of m' independent of m ; but m and m' entering alike into
the preceding radical, they ought to enter similarly into the expression of
13 ; we have therefore
7 being a coefficient independent of m and m' ; therefore
. 1 — m
Thus we see that /3 is split into three factors, the first independent of
m and m' ; the second a function of m' alone ; and the third a like function
of m. We have only now to determine 7.
For that purpose, we shall observe, that if i — n be even ; we have,
supposing m = 0, and m' = 0,
A _ y.U.2....i — n}2 _
= [2. 4. . . . (i — n). (2 i — 1). (2 i — 3). . . . (i + n+ I)}2
7. U- 3. 5....(i — n—1). 1.8.5.... (i + n—l)j«
U.3. 5.... (2 i— !)}«
240 A COMMENTARY ON [SECT. XII. & XIII.
If i — n is odd, we shall have, in retaining only the first power of m,
and m',
_ y.m.m' {1. 2.... (i — n)}* _
_
" [2. 4 ---- (i — n— 1) (2i— 1) (2i — 3)..
_ y.m. m' H. 3. 5.... (i — n). 1.3. 5.... (i + n)} 2
U- 3. 5 ____ (2 i— 1)1 2
The preceding radical becomes, neglecting the squares of m, m',
{r*-2 R r cos.(w-*/)+ R 2}~* + R r. m nr {r*-2r R cos. (»-»') + R 2}~' ; . (f )
If we substitute for cos. (a — ar'), its value in imaginary exponentials,
and if we call c the number whose hyperbolic logarithm is unity, the part
independent of m m', becomes
{r — R.c(-— -OV31"1}"*. [r— B.c^(*-«')v^=I]-^.
The coefficient of
Ri cn(w-OV-l + c_n(w-w')V-l R1
TTTT- - 2 - ' or of rT+ri • cos- n (w — w )
in the developement of this function is
2. 1. 3. 5 ____ (i + n — 1). 1. 3. 5 ____ (i — n — 1)
2. 4. 6 ---- (i + n) 2. 4. 6 ____ (i — n)
This is the value of /3 when i — n is even. Comparing it with that
which in the same case we have already found, we shall have
/I. 3. 5. ...(2i— l)x* i(i— l)....(i — n+ 1)
'" \ 1.2.3 ---- i / ^(i+l)(i+2) ---- (i + n)
When n = 0, we must take only half this coefficient, and then we
have
_ /I. 3. 5 ____ 2i— K2
7 := \ 1. 2. 3 ____ i /
R1
In like manner, the coefficient of - , . , m . m' cos. n (* — •&'} in the
r ' + i
function (f) is
2. 1. 3. 5 ____ (i + n) . 1. 3. 5 ---- (i — n)
2. 4. 6. (i + n — 1) . 2. 4. 6 (i — n — 1) '
this is the coefficient of m m' in the value of /3, when we neglect the
squares of m, m', and when i — n is odd. Comparing this with the va
lue already found, we shall have
> /I- 3. 5 (2i— IK* i(i — l)....(i — n + 1.
VV 1.2.3 i ) ' (i+1) (i + 2) (i + n)'
an expression which is the same as in the case of i — n being even.
If n = 0, we also have
/I. 3. 5.... (2 i- IV
7 \ 1.2.3 i ) '
BOOK I.] NEWTON'S PRINCIPIA. 241
560. From what precedes, we may obtain the general form of functions
Y w of m, V 1 — m 2 . sin. -a, and V 1 — m 2 . cos. «r, which satisfy the
equation of partial differences
0 = - - .
\ dm / 1 — m2
Designating by /3, the coefficient of sin. n &, or of cos. n ^, in th
function Y (1), we shall have
- - r— - -T. .
dm 1 — m2
8 is equal to (I — m2) & multiplied by a rational and entire function of m,
and in this case, by the preceding No., we have
A (n) being an arbitrary constant ; thus the part of Y (i> depending on the
angle n », is
+ B (n' cos. n *?} ;
A (n) and B (n) being two arbitraries. If we make successively in this func
tion, n = 0, n = 1, 11 = 2 . . . n = i ; the sum of all the functions which
thence result, will be the general expression of Y (l), and this expression
will contain 2 i + 1 arbitraries B c°>, A «>, B <•», A ®, B <2>, &c.
Let us now consider a rational and entire function S of the order s,
of the three rectangular coordinates x, y, z. If we represent by R the
distance of the point determined by these coordinates from their origin ;
by 6 the angle formed by R and the axis of x ; and by -a the angle which
the plane of x, y forms with the plane passing through R and the axis of
x ; we shall have
x = Rm;y = R. VI — mz. cos. «; z = R V 1 — m2. sin. «.
Substituting these values in S, and developing this function into sines
and cosines of the angle -a and its multiples, if S is the most general func
tion of the order s, then sin. n w, and cos. n *r, will be multiplied by func
tions of the form
n
(1 _ ms) MA .ms-n + B.m8-"-1 -f C.ms~n-2 + &c.};
thus the part of S, depending on the angle n «, will contain 2 (s — n-f-1)
indeterminate constants. The part of S depending on the angle ^ and its
multiples will contain therefore s (s + 1) indeterminates; the part inde-
VOL. II. Q,
242 A COMMENTARY ON [SECT. XII. & XIII.
pendent of » will contain s + 1, and S will therefore contain (s + 1) *
indeterminate constants.
The function Y(0> + Y(1) + &c. Y (s> contains in like manner (s + 1) *
indeterminate constants, since the function Y (i) contains 2 i + 1 ; we may
therefore put S into a function of this form, and this may be effected as
follows :
From what precedes we shall learn the most general expression of Y (s) ,
we shall take it from S and determine the arbitraries of Y (s) so that the
powers and products of m and V 1 — m 2 of the order s shall disappear
from .the difference S — Y (s) ; this difference will thus become a function
of the order s — 1 which we shall denote by S'. We shall take the most
general expression of Y (s - 1} ; we shall subtract it from S', and determine
the arbitraries of Y^""15 so that the powers and products of m and
V 1 — m 2 of the order s — 1 may disappear from the difference
S' — Y(s-1). Thus proceeding we shall determine the functions Y (s),
Y<s-1), Y(s-2), &c. of which the sum is S,
561. Resume now, the equation of No. 559,
U « =fs . Ri + 2 d R . d m'. d */. Q «.
Suppose R a function of m', -a' and of a parameter a, constant for all
shells of the same density, and variable from one shell to another. The
difference d R being taken on the supposition that m', */ are constant we
shall have
therefore
a • d m' d "'• Q "•
Let R i + 3 be developed into a series of the form
Z'W + Z'W + Z'® + &c,
Z' (i) being whatever i may be, a rational and entire function of m',
^/ 1 _ nv7^. sin. w'} and VI — m' 2. cos. •&', which satisfies the equation
of partial differences
The difference of Z' (i) taken relatively to a, satisfies also this equation,
and consequently it is of the same form ; by the general theorem of 556,
we ought therefore only to consider the term Z' (i> in the developement of
R i + 3, and then we have
BOOK I.] k NEWTON'S PRINCIPIA. 243
When the spheroid is homogeneous and differing but little from a
sphere, we may suppose g = 1, and R = a ( 1 + a y') ; then we have, by
integrating relatively to a
U co = L /Z' «. d m' . d */. Q «.
1 + «
Moreover, if we suppose y' developed into a series of the form
Y'«» + Y/(1> + Y'® + &c.,
Y (i) satisfying the same equation of partial difference as Z' (i) ; we shall have,
neglecting quantities of the order a2, Z'w: = (i + 3). a. ai + 3 Y/(i) ; we
shall therefore have
U ® = a . a4 +3./ Y' «. d m'. d V. Q «.
If we denote by Y (i) what Y' (i) becomes when we change m' and -a' into
m and -a ; we shall have by No. 554,
U« - --L— Y«-
2 i + 1
we therefore have this remarkable result,
4 v Y (i)
(1)
This equation subsisting whatever may be Y' (i) we may conclude ge
nerally that the double integration of the function f Z' (i) d m'. d «•' . Q W
taken from m' = — 1 to m' = 1, and from */ = 0 to •*' = 2 T, only
4 v Z (i)
transforms Z' (i) into =-. — — =- ; Z (i) being what Z' (i) becomes when we
change m' and •*' into m and -a ; we therefore have
4 * , /d Z (i\ .
= - d a;
and the triple integration upon which U (i) depends, reduces to one in
tegration only taken relatively to a, from a = 0 to its value at the surface
of the spheroid.
The equation (1) presents a very simply method of integrating the func
tion f Y (i). Z (i). d m . d 9, from m = — 1 to m = 1, and from -a — 0
to « = 2 it. In fact, the part of Y « depending on the angle n v, is by
what precedes, of the form X {A^ sin. n » + B(n) cos. n *}, X being
equal to
(1 __ m-*)f - {mi-n-i1-11^1.^-1) . m^ — ^ + &c. };
we shall have therefore
Y' (i) = X' {A (n) sin. n *>' + B(n) cos. n *S\ ;
X7 being what X becomes when m is changed into m'. The part of Q (l)
depending on the angle n *», is by the preceding No., y X X' cos. n (*r — »'),
244 A COMMENTARY ON [SECT. XII. & XIII.
or 7 X'. X.£cos. n a. cos. n «/ + sin. n ». sin. n •&'} ; thus that part of the
integral/ Y «. d m . d »•'. Q 0) which depends on the angle n », will be
7 A. sin. n ^./X'2. d m'. d */. sin. n */ {A <n> sin. n »•' + B W cos. n </}
7 X. cos. n w/x' 2. d m'. d -at. cos. n »' JA (n^ sin. n ^ + B (n> cos. n w'}.
Executing the integrations relative to =•', that part becomes
7 X -a [A (n) sin. n * + B (n) cos. n wj./x'2. d m' ;
but in virtue of equation (1), the same part is equal to
4<r
X-T— ; — r . X. f A (n> sin. n w + B (n> cos. n «r?
Now represent by X £A/(n> sin. n » + B/(n> cos. n sr] that part of Z (i)
which depends on the angle n *. This part ought to be combined with
the corresponding part of Y ^ ; because the terms depending on the sines
and cosines of the angle » and its multiples, disappear by integration, in
the function/ Y (i) Z (i) d m . d », integrated from •* — 0 to -a = 2 *; we
shall thus obtain, in regarding only that part of Y (i> which depends on
the angle n w,
/Y W. Z W d m d * =
/X 2. d m . d *{ A W sin. n « + B (n) cos. n *,} {A.' <n> sin. n 9 + B' (n) cos. n *}
B7^)}. A«dm=. .4<r
Supposing therefore successively in the last member n = 0, n = 1,
n = 2 . . . n = i; the sum of all the terms, will be the value of the in
tegral/ Y« ZWdm.dw.
If the spheroid is one of revolution, so that the axis with which the ra
dius R forms the angle w, may be the axis of revolution ; the angle -a will
disappear from the expression of Z (i), which then takes the following
form:
1.3.5. ..2i— 1 nf n i. (i-1) , i.(i-l) (i-2) (i-3> .
W < m 03— — > _ L. m ' ~ 2 -I- — - - — - — - -m1"*— ??c
( 2.2i-l +2.4.2 C'
__ - _
1.2.3. ..i
A (i) being a function of a. Call X W the coefficient of A (i), in this func
tion : the product
/1.3.5...(2i— IK2 ( . i. (i— 1) 1s
( 1.2.3.. .i )-i1-a.(8i-i)-+*c'}'
R'
is by the preceding No., the coefficient of • j in the developement of
the radical
2— 2 R r {m m'+ V 1 — m 2. V
cos.
BOOK I.] NEWTON'S PIUNCIPIA. 245
when we therein suppose m and m' equal to unity. This coefficient is
then equal to 1 ,- we have therefore
I.3.5...(2i-l)( i (i - 1) 1 _
1. 2. 3...i V 2 (2 i — lp J"
that is to say, X W reduces to unity, when m = 1. We have then
_ _
" (i + 3). (2i + Iy da
Relatively to the axis of revolution, m = 1, and consequently,
4 it /d A (i)
therefore if we suppose that relatively to a point placed upon this axis
produced, we have
we shall have the value of V relative to another point placed at the mean
distance from the origin of coordinates, but upon a radius which makes
with the axis of revolution, an angle whose cosine is m ; by multiplying
the terms of this value respectively by X c% X W, X (% &c.
In the case when the spheroid is not of revolution, this method will
give the part of V independent of the angle -a : we shall determine the
other part in this manner. Suppose for the sake of simplicity, the sphe
roid such that it is divided into two equal and similar parts by the equa
tor, whether by the meridian where we fix the origin of the angle &, or
by the meridian which is perpendicular to the former. Then V will be
a function of m 2, sin. 2 w, and cos. 2 •», or which comes to the same, it will
be a function of m2, and of the cosine of the angle 2 -a and its multiples ;
U (l) will therefore be nothing, when i is odd, and in the case when it is
even, the term which depends on the angle 2 n v, will be of the form
C «. (l-mWm'-«» — £"*"|<?-2n1-1)m'-«°-« +&c.}cos. 2 n ,.
^- *— (/& 1 — — Jij J
Relatively to an attracted point placed in the plane of the equator,
where m = 0, that part of V which depends on this term becomes
+ C») _ 1. 3. 5...(i — 2n— 1) __
rri + 1' 2 (i + 2n + l)(i + 2n + 2)...(2i— 1) *
whence it follows that having developed V into a series ordered according
to the cosines of the angle. 2 •& and its multiples, when the point attracted
is situated in the plane of the equator ; to extend this value to any attract
ed point whatever, it will be sufficient to multiply the terms which depend
COS. 2 n nr
on — -r— - — by the junction
Q3
216 A COMMENTARY ON [SECT. XII. & XIII.
_
-1.3.5. ..(i — 2n— I)' M ~2~(2 i — 1)
mi_2n-2+
we shall hence obtain, therefore, the entire value of V, when this value
shall be determined in a series, for the two cases where the part attracted
is situated upon the polar axis produced, and where it is situated in the
plane of the equator; this greatly simplifies the research of this value.
The spheroid which we are considering comprehends the ellipsoid.
Relatively to an attracted point situated upon the polar axis, which we
shall suppose to be the axis of x, by 546, we have b = 0, c — 0, and
then the expression of V of No. 549, is integrable relatively to p. Rela
tively to a point situated in the plane of the equator, we have a = 0, and
the same expression of V still becomes, by known methods, integrable re
latively to q, by making tan. q = t. In the two cases, the integral being
taken relatively to one of these variables in its limits, it then becomes
possible relatively to the other, and we find that M being the mass of
V
the spheroid, the value of ^ is independent of the semi-axis k of the
spheroid perpendicular to the equator, and depends only on the ex-
centricities of the ellipsoid. Multiplying therefore the different terms
V
of the values of ^ relative to these two cases, and reduced into se
ries proceeding according to the powers of - , by the factors above men-
y
tioned, to get the value of -r-=. relative to any attracted point whatever; the
function which thence results will be independent of k, and only depend
on the excentricities ; this furnishes a new demonstration of the theorem
already proved in 550.
If the point attracted is placed in the interior of the spheroid, the at
traction which it undergoes, depends, as we have seen in No. 553, on the
function v (i;, and by the No. cited, we have
r?A Rdm'dV. Q«
TT \l) — • / *L -- •
-J R1-1
an equation which we can put under this form
d a-d m/- d "- Q(i)-
Suppose R2-1 developed into a series of the form
z'(0)+ Z'U)+ Z'(2) + £C.
BOOK I.] NEWTON'S PR1NCIPIA. -. 247
z' (i> satisfying the equation of partial differences,
j / / -
dl(l —
(
if moreover we call z w what z' (i) becomes when we change m' into m, and
v into -0) we shall have by what precedes,
4«r . >dz(i
thus therefore we shall get the expression of V relative to all the shells of
the spheroid which envelope the point attracted. The value of V relative
to shells to which it is interior, we have already shown how to deter
mine.
ON THE FIGURE OF A FLUID HOMOGENEOUS MASS IN EQUILIBRIUM,
AND ENDOWED WITH A ROTATORY MOTION.
562. Having exposed in the preceding Nos. the theory of the attrac
tions of spheroids, we now proceed to consider the figure which they
must assume in virtue of the mutual action of their parts, and the other
forces which act upon them. We shall first seek the figure which satis
fies the equilibrium of a fluid homogeneous mass endowed with a rotatory
motion, and of that problem we shall give a rigorous solution.
Let a, b, c be the rectangular coordinates of any point of the surface of
the mass, and P, Q, R the forces which solicit it parallel to the coordi
nates, the forces being supposed as tending to diminish them. We know
that when the mass is in equilibrium, we have
0 = P. da + Q. db + R. d c;
provided that in estimating the forces P, Q, R, we reckon the centrifugal
force due to the motion of rotation.
To estimate these forces, we shall suppose that the figure of the fluid
mass, is that of the ellipsoid of revolution, whose axis of rotation, is the axis
itself of revolution. If the forces P, Q, R which result from this hypothe
sis, substituted in the preceding equation of equilibrium give the differen
tial equation of the surface of the ellipsoid ; the preceding hypothesis is
legitimate, and the elliptic figure satisfies the equilibrium of the fluid
mass.
Suppose that the axis of a is that also of revolution ; the equation of
the surface of the eUipsoid will be of this form
a2 + m (b2 + c2) = ks;
Ql
248 A COMMENTARY ON [SECT. XII. & XIII.
the origin of the coordinates a, b, c being at the center of the ellipsoid,
k will be the semi-axis of revolution, and if we call M the mass of the el
lipsoid, by 546, we shall have
3 m
g being the density of the fluid. If we make as in 547, m = X 2, we
shall have m = , and consequently
* "*
.
an equation which will give the semi-axis k, when X is known.
Let
B/ = ^*(1 +*2)tan.-'X — X)};
we shall have by 547, regarding only the attraction of the fluid mass
P = A'a; Q = B'b; R = B' c.
If we call g, the centrifugal force at the distance 1, from the axis of
rotation ; this force at the distance V b " + c 2 from the same axis, will
be g V b2 -f- cs: resolving this parallel to the coordinates b, c there will
result in Q the term — g b, and in R the term — g c; thus we shall have,
reckoning all the forces which animate the molecules of the surface,
P = A'a; Q = (B'-g)b; R = (B'— g). c;
the preceding equation of equilibrium, will therefore become
0 = a d a + B ~g (b d b + c d c).
The differential equation of the surface of the ellipsoid is by substitut
in for m its value
,
X
b d b -f- c d c
-
,
= ad + x
by comparing this with the preceding one, we shall have
(1 + >-2)(B'-g) = A'; ........ (1)
if we substitute for A', B' their values, and if we make r^— = q ; we shall
7V £
have
BOOK I.J NEWTON'S PRINCIPIA. 249
determining therefore X by this equation which is independent of the co
ordinates a, b, c, the equation of equilibrium will coincide with the equa
tion of the surface of the ellipsoid ; whence it follows, that the elliptic fi
gure satisfies the equilibrium, at least, when the motion of rotation is such
that the value of X 2 is not imaginary, or when being negative, it is neither
equal to nor greater than unity. The case where X2 is imaginary would
give an imaginary solid; that where X2 = — 1, would give a paraboloid,
and that where X 2 is negative and greater than unity, would give a hy-
perboloid.
563. If we call p the gravity at the surface of the ellipsoid, we shall
have
p = V P 2 + Q 2 + R 2.
In the interior of the ellipsoid, the forces P, Q, R, are proportional to
the coordinates a, b, c ; for we have seen in No. 547, that the attractions
of the ellipsoid, parallel to these coordinates, are respectively proportional
to them, which equally takes place for the centrifugal force resolved pa
rallel to the same coordinates. Hence it follows, that the gravities at dif
ferent points of a radius drawn from the center of the ellipsoid to its sur
face, have parallel directions, and are proportional to the distances from
the center ; so that if we know the gravity at its surface, we shall have
the gravity in the interior of the spheroid.
If in the expression of p, we substitute for P, Q, R, their values given
in the preceding No., we shall have
p = V A'2 a2 + (B'-g)2. (b8 + c2);
whence we derive, in virtue of equation (1) of the preceding No.
p =
-^ ^x -r /v ;
b 2 + c2
but the equation of the surface of the ellipsoid gives -, $• = k2 — a2;
1 -j~ A
we shall therefore have
/ k * + X 2 a~*
AV 1 + x2 '
a is equal to k at the pole, and it is nothing at the equator ; whence it fol
lows, that the gravity at the pole is to the gravity at the equator, as
V 1 + X 2 is to unity, and consequently, as the diameter of the equatoi
is to the polar axis.
Call t the perpendicular at the surface of the ellipsoid, produced to
meet the axis of revolution, we shall have
t = V (1 + X2) (k2 + X'a2) ;
250 A COMMENTARY ON [SECT. XII. & XIII.
wherefore
A't
=
1 + A2
thus gravity is proportional to t.
Let 4- be the complement of the angle which t makes with the axis of
revolution ; 4 will be the latitude of the point of the surface, which we
are considering, and by the nature of the ellipse, we shall have
V 1 + A2cos.2<4/
we therefore shall have
_ A'k
V I + X2. cos.2 4'
and substituting for A' its value, we shall get
4«rg.k.(l + X2). (X — tan.-'X) . .
X3 V 1 + X2. cos.2 4
this equation gives the relation between gravity and the latitude ; but we
must determine the constants which it contains.
Let T be the number of seconds in which the fluid mass will effect a
revolution ; the centrifugal force at the distance 1 from the axis of revo-
4 7T2
lution, will be equal to -™y ; we therefore have
g 12 g2
q==f*..e- 4*fT25
which gives
12. cr2
4 K P —
q. 1
The radius of curvature of the elliptic meridian is
(l+A2)k ,
( 1 + X 2 cos. 2 4<) 2
calling therefore c the length of a degree at the latitude -4y\ve shall have
— i — + X '* =• = 200 c.
/ 1 i •> 2 2 J/\ f
This equation combined with the preceding one, gives
4?rg (1 + Xj^_ _ 20() c ^] + X2 cos.s ^ l2^.
V 1 + A 2 cos. 2 4/ q J-
thus we shall have
A — tan. -'A 12 v
q"
Let 1 be the length of the simple pendulum which oscillates seconds ;
p = 200 C(l + X2COS.24) -3 .-7TT5.
A Cl JL
BOOK!.] NEWTON'S PRINCIPIA. 231
from dynamics it results that p = -r2 1 (see§X.) ; comparing these two
expressions of p, we get
- 2400 c (A — tan.-1*.) (1 + X2cos.2^) .
*1T2A3
this equation and equation (2) of the preceding No. will give the values
of q and X by means of the length 1 of the seconds' pendulum, and the
length c of the degree of the meridian, both being observed at the lati
tude -^.
Suppose 4> = 50°, these equations will give
. 800c .i
• *1T2 4 WlT
observations give, as we shall see hereafter,
c = 100000;!= 0.741608;
moreover we have T = 99727 ; we shall thus obtain
q = 0.00344957 ; X 2 = 0.00868767.
The ratio of the axis of the equator to the polar axis, being V 1 + X 2,
it becomes in this case 1.00433441 ; these two axes are very nearly in
the ratio of 231.7 to 230.7, and by what precedes, the gravities at the
pole and at the equator are in the same ratio.
We shall have the semi polar axis k, by means of the equation
200 c (1 + ^2)*' i
*(1 + X2) ~^~{i
which gives
k = 6352534.
To get the attraction of a sphere, whose radius is k. and density any
whatever ; we shall observe that a sphere, having the radius k and density
f, acts upon a point placed at its surface, with a force equal to f is g . k,
and consequently, in virtue of equation (3) equal to rTT>
o ( 1 -j- A j (X — tan. X)
/ o
or to p (l — -_ x2 + &c.V or finally to 0.998697. p, p being the gravi
ty upon the parallel of 50°. Hence it is easy to obtain the attractive force
of a sphere of any radius and density whatever, upon a point placed with
in or without it.
564. If the equation (2) of No. 562, were susceptible of many real
roots, many figures of equilibrium might result from the same motion of
rotation ; let us examine therefore whether this equation has several real
252 A COMMENTARY ON [SECT. XII. & XIII.
0X1 2 Q X 3
roots. For that purpose, call <p the function — - - 5—^ -- tan.-lX,
y "I* o A
which being equated to zero, produces the equation (2). It is easy to see,
that by making X increase from zero to infinity, the expression of <p begins
and ends by being positive ; thus, by imagining a curve whose abscissa is
X and ordinate p, this curve will cut its axis when X = 0 ; the ordinates
will afterwards be positive and increasing ; when arrived at their maxi
mum, they will decrease; the curve will cut the axis a second time at a
point which will determine the value of X corresponding to the state of
equilibrium of the fluid mass; the ordinates will then be negative, and
since they are positive when X = oo ; the curve necessarily cuts the axis
a third time, which determines a second value of X which satisfies the
equilibrium. Thus we see, that for one and the same value of q, or for
one given motion of rotation, there are several figures for which the
equilibrium may subsist.
To determine the number of these figures, we shall observe, that we
have
_ 6 XgdXJq X4 + (10 q — 6) X*+ 9 q}
(3 X2+9)2. (1 + X2)
The supposition of d <p = 0, gives
0 = qX* + (10 q— 6) X2+ 9 q;
whence we derive, considering only the positive values of X
These values of X determine the maxima and minima of the ordinate <p ;
there being only two similar ordinates on the side of positive abscissas, on
that side the curve cuts its axis in three points, one of them being the
origin ; thus, the number of figures which satisfy the equilibrium is reduc
ed to two.
The curve on the side of negative abscissas being exactly the same as
on the side of positive abscissas ; it cuts its axis on each side in corre
sponding points equidistant from the origin of coordinates ; the negative
values of X which satisfy the equilibrium, are therefore, as to the sign
taken, the same as the positive values ; which gives the same elliptic fi
gures, since the square of X only enters the determination of these figures ;
it is useless therefore to consider the curve on the side of negative ab
scissas.
If we suppose q very small, which takes place for the earth, we may
satisfy equation (2) of 562, in the two hypotheses of X 2 being very small,
BOOK L] NEWTON'S PRINCIPIA. 253
and of X 2 being very great. In the first, by the preceding No., we
have
To get the value of X2 in the second hypothesis, we shall observe that
CT
then tan." 1 X differs very little from £ *, so that if we suppose X = - — «,
a will be a very small angle of which the tangent is - ; we shall there
fore have, p. 27. Vol. I.
1
a= x
and consequently
equation (2) of No. 562, will thus become^
9X+2q.X*__ ff 1 . j_
9 + 3X2 "2 X^3X3
whence by the reversion of series we get
3cr 8 4 q /. 64 N
X = - ---- -- M 1 — - — 2} + &c.
4 T T \ 3 cr v
-
4 q T T \ 3 cr
= 2.356195. -L — 2.546479 — 1.478885 q + &c.
We have seen in the preceding No., that relatively to the earth,
q = 0.00344957 ; this value of q substituted in the preceding expression,
gives X = 680.49. Thus the ratio of the two axes equatorial and polar,
a ratio which is equal to V 1 + X 2, is in the case of a very thin spheroid,
equal to 680.49.
The value of q has a limit beyond which the equilibrium is impossible,
the figure being elliptic. Suppose, in fact, that the curve cuts its axis
only at its origin, and that in the other points it only touches; at the
point of contact we shall have <p = 0, and d <p = 0 ; the value of p will
never therefore be negative on the side of positive abscissas, which are
the only ones we shall here consider. The value of q determined by the
two equations p = 0, d p = 0, will therefore be the limit of those with which
the equilibrium can take place, so that a greater value will render the
equilibrium impossible ; for q being supposed to increase by f, the func-
2 f X 3
tion <p increases by the term jr — ^ ; thus, the value of <p correspond-
*s "^ o A
ing to q, being never negative, whatever X may be, the same function cor
responding to q + fj is constantly positive, and can never become no-
254 A COMMENTARY ON [SECT. XII. & XIII.
thing ; the equilibrium is then therefore impossible. It results also from
this analysis, that there is only one real and positive value of q, which
would satisfy the two equations <p = 0, and d <p — 0. These equations
give
7 X 5 + 30 X 3 + 27 X
= -
_
= (1 + X2) (9 + X2)
7 X 5 + 30 X 3 +
(1 + x2)(3 + X*)
The value of X which satisfies this last equation is X = 2.5292 ; whence
we get q = 0.337007 ; the quantity V 1 -J- X 2, which expresses the ra
tio of the equatorial axis to the polar axis, is in this case equal to 2.7197.
The value of q relatively to the earth is equal to 0.00344957. This
value corresponds to a time of rotation of 0.99727 days ; but we have
generally q = r**— » so that relatively to masses of the same density, q is
o 5
proportional to the centrifugal force g of the rotatory motion, and conse
quently in the inverse ratio of the square of the time of rotation ; whence
it follows, that relatively to a mass of the same density as the earth, the
time of rotation which answers to q = 0.337007, is 0.10090 days. Whence
result these two theorems.
" Every homogeneous fluid mass of a density equal to the mean density
of the earth, cannot be in equilibrium having an elliptic figure, if the time
of its rotation is less than 0.10090 days. If this time be greater, there
will be always two elliptic figures and no more which satisfy the equili
brium."
" If the density of the fluid mass is different from that of the earth ; we
shall have the time of rotation in which the equilibrium ceases to be pos
sible under an elliptic figure, by multiplying 0.10090 days by the square
root of the ratio of the mean density of the earth to that of the fluid
mass."
This relatively to a fluid mass, whose density is only a fourth part of
that of the earth, which nearly is the case with the sun, this time will be
0.20184 days; and if the density of the earth supposed fluid and homo
geneous were about 98 times less than its actual density, the figure which
it ought to take to satisfy its actual motion of rotation, would be the limit
of all the elliptic figures with which the equilibrium can subsist. The
density of Jupiter being about five times less than that of the earth, and
the time of its rotation being 0.41377 days; we see that this duration is
in the limits of those of equilibrium.
BOOK I.] NEWTON'S PRINCIPIA.
It may be thought that the limit of q, is that where the fluid would be
gin to fly off by reason of a too rapid motion of rotation ; but it is easy to
be convinced of the contrary, by observing that by 563, the gravity at the
equator of the ellipsoid is to that at the pole in the ratio of the polar axis
to that of the equator, a ratio which in this case, is that of 1 to 2.7197 ;
the equilibrium ceases therefore to be possible, because with a motion of
rotation more rapid, it is impossible to give to the fluid mass, an elliptic
fio-ure such that the resultant of its attraction and of the centrifugal force,
may be perpendicular to the surface.
Hitherto we have supposed X 2 positive, which gives the spheroids flat
tened towards the poles ; let us now examine whether the equilibrium can
subsist with a figure lengthened towards the poles, or with a prolate sphe
roid. Let X 2 = — X/2 ; X' 2 must be positive and less than unity, otherwise,
the ellipsoid will be changed into a hyperboloid. The preceding value
of d p gives
x.X2dx£qX4+ (10 q — 6) Xg + 9 qj _
^ ~ J ~ (1 + X2) (9 + 3 X2)2
the integral being taken from X = 0. Substituting for X its value + X' V - 1,
we shall have
-— - -
l'J ' (l _x/2) (9 —3 X'2)
but it is evident that the elements of this last integral are all of the same
sign from X/2 = 0, to X/2 = 1 ; the function p can therefore never be
come nothing in this interval. Thus then the equilibrium cannot subsist
in the case of the prolate spheroid.
565. If the motion of rotation primitively impressed upon the fluid
mass, is more rapid than that which belongs to the limit of q, we must
not thence infer that it cannot be in equilibrium with an elliptic figure ;
for we may conceive, that by flattening it more and more, it will take a
rotatory motion less and less rapid ; supposing therefore that there exists,
as in the case of all known fluids, a force of tenacity between its mole
cules, this mass, after a great number of oscillations, may at length arrive
at a rotatory motion, comprised within the limits of equilibrium, and may
continue in that state. But this possibility it would also be interesting to
verify ; and it would be equally interesting to know whether there would
not be many possible states of equilibrium ; for what we have already de
monstrated upon the possibility of two states of equilibrium, correspond
ing to one motion of rotation, does not infer the possibility of two states
of equilibrium corresponding to one primitive force ; because the two
256 A COMMENTARY ON [SECT. XII. & XIII.
states of equilibrium relative to one motion of rotation, require two pri
mitive forces, either different in quantity or differently applied.
Consider therefore a fluid mass agitated primitively by any forces what
ever,- and then left to itself, and to the mutual attractions of all its parts.
If through the center of gravity of this mass supposed immoveable, we
conceive a plane relatively to which the sum of the areas described upon
this plane, by each molecule, multiplied respectively by the correspond
ing molecules, is a maximum at the origin of motion ; this plane will
always have this property, whatever may be the manner in which the
molecules act upon one another, whether by their tenacity, by their attrac
tion, and their mutual collision, even in the very case where there is finite
loss of motion in an instant of time ; thus, when after a great number of
oscillations, the fluid mass shall take a uniform rotatory motion about a
fixed axis, this axis shall be perpendicular to the plane above-mentioned,
which will be that of the equator, and the motion of rotation will be such
that the sum of the areas described during the instant d t, by the mole
cules projected upon this plane, will be the same as at the origin of mo
tion ; we shall denote by E d t this last sum.
We shall here observe, that the axis in question, is that relatively to
which the sum of the moments of the primitive forces of the system was a
maximum. It retains this property during the motion of the system, and
finally becomes the axis of rotation ; for what is above asserted as to the
plane of the maximum of projected areas, equally applies to the axis of the
greatest moment of forces ; since the elementary area described by the pro
jection of the radius-vector of a body upon a plane, and multiplied by its
mass, is evidently proportional to the moment of the finite force of this
body relatively to the axis perpendicular to this plane.
Let, as above, g be the centrifugal force due to the rotatory motion at
the distance 1 from the axis; V g will be the angular velocity of rotation
(p. 166. Vol. I.) ; then call k the semi-axis of rotation of the fluid mass,
and k V 1 + A z the semi-axis of its equator. It is easy to show that
the sum of the areas described during the instant d t, by all the molecules
projected upon the plane of the equator and multiplied respectively by the
corresponding molecules, is
1(1 + A*)2.k5dt Vg
we shall therefore have
BOOK I.] NEWTON'S PRINCIPIA. 257
Then calling M, the fluid mass, we shall have
$*k>ff(l + X2) = M;
the quantity r^— > which we have called q, in No. 562, thus becomes
q' (1 + X £)~^ denoting by q' the function 2^J_JL£lf The equa
tion of the same No. becomes
9 + 3X2
This equation will determine X ; we shall then have k by means of the
preceding expression of M.
Call <p the function
9 + 3X2 tan' ~~l X>
which, by the condition of equilibrium, ought to be equal to zero : this
equation begins by being positive, when X is very small, and ends by being
negative, when X is infinite ; there exists therefore between X = 0, and
X = infinity, a value of X which renders this function nothing, and conse
quently, there is always, whatever q' may be, an elliptic figure, with which
the fluid mass may be in equilibrium.
The value of <p may be put under this integral form
/X4dx{^-+ 18 q' — fq'X2 + 18(1 + X 2)
<f> = 2 I L-
(9 + 3 X2)2 (1 + X2)*
When it becomes nothing the function
—?' + 18q'-fq'X2 + 18(1 + *2)f],
has already passed through zero to become negative ; but from the in
stant when this function begins to be negative, it continues to be so as X
27 q'
increases, because the positive part f> + 18 q' decreases whilst the ne-
X
gative part — {q' X2 + 18 (1 + X2)S} increases; the function p cannot
therefore twice become nothing ; whence it follows, that there is but one
real and positive value of X which satisfies the equation of equilibrium,
and consequently, the fluid can be in equilibrium with one elliptic figure
only.
Vor.. IT. R
258 A COMMENTARY ON [SECT. XII. & XIII.
ON THE FIGURE OF A SPHEROID DIFFERING VERY LITTLE FROM A SPHERE,
AND COVERED WITH A SHELL OF FLUID IN EQUILIBRIUM.
566. We have already discussed the equilibrium of a homogeneous
fluid mass, and we have found that the elliptic figure satisfies this equili
brium; but in order to get a complete solution of the problem, \ve must
determine a priori all the figures of equilibrium, or we must be certified
that the elliptic is the only figure which will fulfil these conditions; be
sides, it is very probable that the celestial bodies have not homogeneous
masses, and that they are denser towards the center than at the surface.
In the research, therefore, of their figure, we must not rest satisfied with
the case of homogeneity ; but then this research presents great difficul
ties. Happily it is simplified by the consideration of the little difference
which exists between the spherical figure and those of the planets and
satellites; by which we are permitted to neglect the square of this differ
ence, and of the quantities depending on it. Notwithstanding, the research
of the figure of the planets is still very complex. To treat it with the
greatest generality, we proceed to consider the equilibrium of a fluid mass
which covers a body formed of shells of variable density, endowed with
a rotatory motion, and sollicited by the attraction of other bodies. For
that purpose, we proceed to recapitulate the laws of equilibrium of fluids,
as laid down in works upon hydrostatics.
If we name g the density of a fluid molecule, II the pressure it sustains,
F, F', F", &c. the forces which act upon it, and d f, d f ', d f " the ele
ments of the directions of these forces; then the general equation of the
equilibrium of the fluid mass will be
— - F d f + F' d f + F" d f " + &c.
S
Suppose that the second member of this equation is an exact difference;
designating by d p this difference, g will necessarily be a function of n and
of <p : the integral of this equation will give <p in a function of n ; we may
therefore reduce to a function of n only, from which we can obtain n in
a function of p ; thus, relatively to shells of a given constant density, we
shall have d n = 0, and consequently
0 = F d f + F d f ' + F" d f" + &c. ;
an equation which indicates the tangential force at the surface of those
shells is nothing, and consequently, that the resultant of all the forces
F, F', F", &c. is perpendicular to this surface ; so that the shells are
spherical.
BOOK I.] NEWTON'S PUINCIPIA. 259
The pressure n being nothing at the exterior surface, § must there be
constant, and the resultant of all the forces which animate each molecule
of the surface is perpendicular to it. This resultant is wnat we call gravi
ty. The conditions of equilibrium of a fluid mass, are therefore 1st, that
the direction of gravity be perpendicular to each point of the exterior sur
face : 2dly, that in the interior of the mass the directions of the gravity of
each molecule be perpendicular to the surface of the shells of a constant
density. Since we may take, in the interior of a homogeneous mass, such
shells as we wish for shells of a constant density, the second of two pre
ceding conditions of equilibrium, is always satisfied, and it is sufficient for
the equilibrium that the first should be fulfilled ; that is to say, that the
resultant of all the forces which animate each molecule of the exterior
surface should be perpendicular to the surface.
567. In the theory of the figure of the celestial bodies, the forces F, F',
F", &c. are produced by the attraction of their molecules, by the centrifu
gal force due to their motion of rotation, and by the attraction of distant
bodies. It is easy to be certified that the difference F d f + F' d f + &c.
is there exact ; but we shall clearly perceive that, by the analysis which
we are about to make of these different forces, in determining that part of
the integralt/(F d f + F' d f ' -f &c.) which is relative to each of them.
If we call d M any molecule of the spheroid, and f its distance from the
point attracted, its action upon this latter will be — ^- . Multiplying this
action by the element of its direction, which is — d f, since it tends to
diminish f, we shall have, relatively to the action of the molecule d M,
/F d f = -p- ; whence it follows that that part of the integral /(F d f
+ F d f + &c.), which depends on the attraction of the molecules of
the spheroid, is equal to the sum of all these molecules divided by their
respective distances from the molecule attracted. We shall represent this
sum by V, as we have already done.
We propose, in the theory of the figure of the planets, to determine
the laws of the equilibrium of all their parts, about their common center of
gravity; we must, therefore, transfer into a contrary direction to the mole
cule attracted, all the forces by which this center is animated in virtue of
the reciprocal action of all the parts of the spheroid; but we know
that, by the property of this center, the resultant of all the actions upon
tliis point is nothing. To get, therefore, the total effect of the attraction
R 2
260 A COMMENTARY ON [SECT. XII. & XIII.
of the spheroid upon the molecules attracted, \ve have nothing to add
to V.
To determine the effect of the centrifugal force, we shall suppose the
position of the molecule determined by the three rectangular coordinates
x', y', z', whose origin we fix at the center of gravity of the spheroid.
We shall then suppose that the axis of x7 is the axis of rotation, and that
g expresses the centrifugal force due to the velocity of rotation at the dis
tance I from the axis. This force will be nothing in the direction of x'
and equal to g y' and g z' in the direction of y' and of z' ; multiplying,
therefore, these two last forces respectively by the elements d y7, d z' of
their directions, we shall have ^ g (y' 8 + z' 2) for that part of the integral
f (F d f + F' d f ' + &c.), which is due to the centrifugal force of the
rotatory motion.
If we call, as above, r the distance of the molecule attracted from the
center of gravity of the spheroid, 6 the angle which the radius r forms with
the axis of x', and •* the angle the plane which passes through the axis
of x', and through the molecule, forms with the plane of x', y' ; finally, if
we make cos. 6 = m, we shall have
x' = r m ; y' = r V 1 — m 2 . cos. -a ; z' = r V 1 — m z . sin. -a ;
whence we get
ig(y/2 + z'2) = *g*'(l— m')-
We shall put this last quantity under the following form :
4gr« — igr«(m« — i)
to assimilate its terms to those of the expression V which are given in No.
559; that is to say, to give them the property of satisfying the equation of
partial differences
in which Y (i) is a rational and entire function of m, V 1 — m * . cos. *
and VI m2 sin. « of the degree i ; for it is clear that each of the two
terms £ g r * and — \ g r 2 (m2 — £) satisfies for Y «, the preceding
equation.
It remains now for us to determine that part of the integral
/"(F d f + F7 d f' + &c.) which results from the action of distant bodies.
Let S be the mass of one of these bodies, f its distance from the molecule
attracted, and s its distance from the center of gravity of the spheroid.
Multiplying its action by the element— d f of its direction, and then inte-
BOOK I.] NEWTON'S PRINCIPIA. 261
c
grating we shall have -TT-. This is not the entire part of the integral
/(F d f + F' d f ' + &c.) which is due to the action of S; we have still
to transfer, in a contrary direction to the molecule, the action of this body
upon the center of gravity of the spheroid. For that purpose, call v the
angle which s forms with the axis of x', and 4< the angle which the plane
passing through this star and through the body S, makes with the plane of
S
x', y'. The action of — ^ of this body upon the center of gravity of the
spheroid, resolved parallel to the axes of x', y', z', will produce the three
following forms :
S S . S .
— g cos. v; — - sin. v cos. 4; •— «• sin. v sin. 4-.
s^ sz s 2
Transferring them in a contrary direction to the molecule attracted,
which amounts toprefixing to them the sign — , then multiplying them by
the elements d x', d y', d z', of their directions, and integrating them, the
sum of the integrals will be
g
-- j- .£x' cos. v + y' sin. v. cos. 41 + z' sin. v sin. -^\ + const. ;
the entire part of the integral /(F d f + F' d f ' + &c.), due to the ac
tion of the body S, will therefore be
S S
-f -- -^i*' c°s. v + y' sin. v cos. 4/ + z' sin. v sin. -^} + const. ;
and since this quantity ought to be nothing relatively to the center of gra
vity of the spheroid, which we suppose immoveable, and that relatively to
this point, f becomes s, and x', y', z', are nothing, we shall have
const. = — — .
s
However, f is equal to
J(s cos. v — x') 2 + (s. sin. v cos. ^ — y') 2 + (s sin. v sin. -4, — z') «}£;
which gives, by substituting for x', y', z', their preceding values
S_= S _
^s * — 2s r£cos. v cos. 6 + sin. v sin. 6 cos. (<*~^~-^)~+~x*}'
If we reduce this function into a series descending relatively to powers
of s, and if we thus represent the series,
we shall have generally by 56 1 and 562,
L3.5..(2i-l) f i(i-l) , i(i-l)(i-2)(l-8) ) ;
1.2.3 ..... i I 2(2 i— If h2.4(2i— ])(2l=3)a I ;
262 A COMMENTARY ON [SECT. XII. .& XIII.
3 being equal to cos. v cos. d + sin. v sin. 6 . cos. (^ — 4) > it is evident
that by 553, we have
0 =
so that the terms of the preceding have this property, common with those
of V. This being shown, we have
s s s
-TT ---- r(x' cos. v + y' sin. v cos. 4> + z' SU1- v sin> "^)
I S S i
p (2) +7 P (3) +7^ p w + &c
If there were other bodies S', S", &c. ; denoting by s', v', 4/'» P' (i) ; s",
v", 4/', P" W, &c. what we have called s, v, 4/, P (i), relatively to the body
S, we shall have the parts of the integral /(F d f + F d P + &c.) due
to their action, by marking with one, two, &c. dashes, the letters s, v, 4^
and P in the preceding expression of that part of this integral, which is
due to the action of S.
If we collect all the parts of this integral, and make
J-=aZ«>;
&C.
a being a very small coefficient, because the condition that the spheroid is
very little different from a sphere, requires that the forces which produce
this difference should themselves be very small ; we shall have
/(Fdf + Fdf + &c.) = V + ar« {Z«>+ Z®+ rZ®+ r'Z^ + &c.{
Z w satisfying, whatever i maybe, in the equation of partial differences
d 8 Z
m -,—
dm
0 - JJ dm J .A q~ x + i (i + J) Z<«.
\~ ~~d~5T~ J[. 1— m2
The general equation of equilibrium will therefore be
f^JL= V + a r2 {Z(0> + Z<2> -f- r Z® r2 Z^ + &c.} . (1)
o
If the extraneous bodies are very distant from the spheroid, we may ne
glect the quantities r * Z (3), r 4 Z (4>, &c., because the different terms of these
quantities being divided respectively by s 4, s3, &c. s/4, s'3, &c. these terms
become very small when s, s', &c. are very great compared with r. This
BOOK I.] NEWTON'S PRINCIPIA. 263
case subsists for the planets and satellites with the exception of Saturn,
whose ring is too near his surface for us to neglect the preceding terms.
In the theory of the figure of that planet, we must therefore prolong the
second member of equation (1), which possesses the advantage of forming
a series always convergent; and since then the number of corpuscles ex
terior to the spheroid is infinite, the values of Z<°>, Z®, &c. are given in
definite integrals, depending on the figure and interior constitution of the
ring of Saturn.
568. The spheroid may be entirely fluid ; it may be formed of a solid
nucleus covered by a fluid. In both cases the equation (1) of the preced
ing No. will determine the figure of the shells of the fluid part, by con
sidering, that since n must be a function of f, the second member of this
equation must be constant for the exterior surface, and for that of the
shells in equilibrium, and can only vary from one shell to another.
The two preceding cases reduce to one when the spheroid is homoge
neous ; for it is indifferent as to the equilibrium whether it is entirely
fluid, or contains an interior solid nucleus. It is sufficient by No. 556, that
at the exterior surface we have
constant = V + a r2 [Z^+ Z®+ r Z® + £c.}.
If we substitute in this equation for V its value given by formula (3) of
No. 555, and if we observe that by No. 556, Y (0) disappears by taking for
a the radius of a sphere of the same volume as the spheroid, and that
Y (l' is nothing when we fix the origin of coordinates at the center of the
spheroid; we shall have
constant =±^l+^Li{J_YB, + JL. Y» +j5f.Y«+ &c.}
+ a r 2 [Z !0> + Z (2> + r Z ® + r 2 Z « + &c.}
Substituting in the equation of the surface of the spheroid for r its value
at the surface 1 + a y, or
a + a a £Y(2) + Y<3> + Y'-> + &c.}
which gives
const. = -^a* ' 8<7a* {5Y(2) + -f-Y(3) +4YW + &C'l
+ a a* {ZW + Z^ + a Z® + a2 Z« + &c.}
We shall determine the arbitrary constant of the first member of this
equation, by means of this equation,
const. = — « a 2 + « a 8 Z w> ;
0
we shall then have by comparing like functions, that is to say, such as are
subject to the same equation of partial differences,
R l
264 A COMMENTARY ON [SECT. XII. & XIII.
i being greater than unity. The preceding equation may be put under the
form
the integral being taken from r = 0 to r = a. The radius a (1 — ay)
of the surface of the spheroid will hence become
We may put this equation under a finite form, by considering that we
have by the preceding No.
so that the integraiy* d r JZ ^ + r Z 3} + &c.} is easily found by known
methods.
569. The equation (1) of 567 not only has the advantage of showing the
figure of the spheroid, but also that of giving 'by differentiation the law of
gravity at its surface ; for it is evident that the second member of this
equation being the integral of the sum of all the forces with which each
molecule is animated, multiplied by the elements of their respective direc
tions, we shall have that part of the resultant which acts along the radius
r, by differentiating the second member relatively to r; thus calling p
the force by which a molecule of the surface is sollicited towards the center
of gravity of the spheroid, we shall have
p = — (^) — ~ d {r8 Z<°> + r2 Z® + r3 Z® + r4 Z<« + &c.|.
If we substitute in this equation for — (—. — \, its value at the surface
2 V
— it a + —— , given by equation (2) of No. 554, and for V, its value given
o *& n
by equation (1) of No. 567; we shall have
p = * a _ a a {z® + a Z(3> a 2 Z<4>
• > £
&c.} (3)
BOOK L] NEWTON'S PRINCIPIA. 265
r must be changed into a after the differentiations in the second mem
ber of this equation, which by the preceding No. may always be reduced
to a finite function.
p does not represent exactly gravity, but only that part of it which is
directed towards the center of gravity of the spheroid, by supposing it re
solved into two forces, one of which is perpendicular to the radius r, and
the other p is directed along this radius. The first of these two forces is
evidently a small quantity of the order a ; denoting it therefore by a 7,
gravity will be equal to Vp2 -f- «2 72, a quantity which, neglecting the
terms of the order a 2, reduces to p. We may thus consider p as express
ing gravity at the surface of the spheroid, so that the equations (2) and
(3) of the preceding No. and of this, determine both the figure of ho
mogeneous spheroids in equilibrium, and the law of gravity at their
surfaces ; they contain the complete theory of the equilibrium of these
spheroids, on the supposition that they differ very little from the sphere.
If the extraneous bodies S, S', &c. are nothing, and therefore the
spheroid is only sollicited by the attraction of its molecules, and the cen
trifugal force of its rotatory motion, which is the case of the Earth and
primary planets with the exception of Saturn, when we only regard the
permanent state of their figures ; then designating by a p, the ratio of
the centrifugal force to gravity at the equator, a ratio which is very nearly
equal to—-, the density of the spheroid being taken for unity; we shall
find,
the spheroid is then therefore an ellipsoid of revolution, upon which in
crements of gravity, and decrements of the radii, from the equator to
the poles, are very nearly proportional to the square of the sine of the
latitude, m being to quantities of the order a, equal to this sine.
a, by what precedes, is the radius of a sphere, equal in solidity to the
spheroid ; gravity at the surface of this sphere will be f v a ; thus we shall
have the point of the surface of the spheroid, where gravity is the same as
at the surface of the sphere, by determining m by the equation
0 =-_§ + f (m« — i)j
which gives
266 A COMMENTARY ON [SECT. XII. & XIII.
570. The preceding analysis conducts us to the figure of a homoge
neous fluid mass in equilibrium, without employing other hypotheses than
that of a figure diifering very little from the sphere : it also shows that
the elliptic figure which satisfies this equilibrium, is the only figure
which does it. But as the expansion of the radius of the spheroid into
a series of the form a [I + a Y(°) + a Y(1) + &c.} may give rise to some
difficulties, we proceed to demonstrate directly, and independently of this
expansion, that the elliptic figure is the only figure of the equilibrium of
a homogeneous fluid mass endowed with a rotatory motion ; which by con
firming the results of the preceding analysis, will at the same time serve
to remove any doubts we may entertain against the generality of this ana
lysis.
First suppose the spheroid one of revolution, and that its radius is a
(1 + a y), y being a function of m, or of the cosine of the angle 6 which this
radius makes with the axis of revolution. If we call f any straight line
drawn from the extremity of this radius in the interior 'of the spheroid ; p
the complement of the angle which this straight line makes with the plane
which passes through the radius a ( 1 + ay) and through the axis of revolu
tion; q the angle made by the projection of f upon this plane and by the
radius ; finally, if we call V the sum of all the molecules of the spheroid,
divided by their distances from the molecules placed at the extremity of
the radius a (1 + a y) ; each molecule being equal to f 2 d f. d p. d q .
sin. p, we shall have
V = i/f/2dp.dq.sin. p,
f being what f becomes at its quitting the spheroid. We must now de
termine f in terms of p and q.
For that purpose, we shall observe that if we call 4', the value of 6 rela
tive to this point of exit, and a (1 + ay'), the corresponding radius of the
spheroid, y' being a similar function of cos. 6' or of m' that y is of m ; it
is easily seen that the cosine of the angle formed by the two sti'aight lines
f and a ( 1 + a y) is equal to sin. p . cos. q ; and therefore that in the
triangle formed by the three straight lines f, a ( 1 + ay) and a ( 1 + a y')
we have
a*(l + ay') z = i'* — 2af'(l + a y) sin. p . cos. q + a2(l +«y)2.
This equation gives for f 2 twa values ; but one of them being of the
order a2 is nothing when we neglect the quantities of that order; the
other becomes
f/2 = 4 a 2 sin. 2 p cos. 2 q (1 + 2 ay) -f- 4 a a2 (y' — y) ;
which gives
BOOK I.] NEWTON'S PRINCIPIA, 267
V - 2 a*/dp clq sin, p {(1 + 2 ay) sin.2p cos.2 q+ « (y — y)J.
It is evident that the integrals must be taken from p = 0, to p = *, and
from q = — £ * to q = £ T ; we shall therefore have
V = f «r a - — § « ?r a 2 y -f- 2 « a 2y*d p . d q . y'sin. p .
y' being a function of cos. S'} we must determine this cosine in a function
of p and q; we may therefore in this determination neglect the quantities
of the order a, since y' is already multiplied by a ; hence we easily find
a cos. 6' = (a — P sin. p cos. q) cos. 6 + f ' sin. p . sin. q . sin. 0 ;
whence we derive, substituting for P its value 2 a sin. p cos. q,
in' = m cos. 2 p — sin. 2 p cos. (2 q + 6).
Here we must observe, relatively to the integral f y' d p . d q . sin. p,
taken relatively to q from 2q = — * to 2 q = '< that the result would
be the same, if this integral were taken from 2 q =r — 0 to 2 q = 2 «• — 0,
because the values of m', and consequently of y' are the same from 2 q =
9 to 2 q = — 6 as from 2 q = «r to 2 q = 2 it — 6 ; supposing there
fore 2 q + 6 = q', which gives
m' = m cos. 2 p — sin. 2 p cos. q' ;
we shall have
V = f TT a 2 — | « cr a 2 y -f a a 2/y' d p d q' sin. p ;
the integrals being taken from p = 0 to p = * and from q' = 0 to q' =
2cr.
Now if we denote by a2 N the integral of all the forces extrinsic to the
attraction of the spheroid, and multiplied by the elements of their direc
tions ; by 568 we shall have in the case of equilibrium
constant = V + a2 N,
and substituting for V its value, we shall have
const. = £ a * • y — a fy' d p . d q' sin. p — N ;
an equation which is evidently but .the equation of equilibrium of No. 568,
presented under another form. This equation being linear, it thence results
that if any number i of radii a (1 + a y), a (1 + a v), and satisfy it; the
radius a { +— (y + v + &c.)} will also satisfy it.
1
Suppose that the extraneous forces are reduced to the centrifugal force
due to the rotation, and call g this force at the distance 1 from the axis of
rotation; we shall have, by 567, N ='£ g (1 — m9) ; the equation of
equilibrium will therefore be
const. = | a v y — a/y d p d q' sin. p — 2 g (1 — m8).
Differentiating three times successively, relatively to m, and observing
that (• — — ^ = cos. " p, in virtue of the equation
\d m /
268 A COMMENTARY ON [SECT. XII. Sc XIII.
m' = m cos. 2 p — sin. 2 p cos. q' j
we shall have
m
— / d p d q' sin. p cos.
J
but we have yd p d q' sin. p cos.6 p = ~j we may therefore put the
preceding equation under this form,
0=/d p d q sin. p co, • p { J (££) - (££) } .
This equation subsists, whatever m may be; but it is evident, that
amongst all the values between m = — 1 and m = 1, there is one which
we shah1 designate by h, and which is such that, abstraction being made
•t q
of the sign, each of the values of (-, — ~3\ will not exceed that which is re
lative to h ; denoting therefore by H, this latter value, we shall have
0 = / d p d q' sin. p cos. B p { I H - (fjlZl) } .
1 q /
The quantity £ H — (-^ — *r*\ has evidently the same sign as H, and
the factor sin. p . cos. 6 p, is constantly positive in the whole extent of the
integral; the elements of this integral have, therefore, all of them the
same sign as H ; whence it follows that the entire integral cannot be no
thing, at least H cannot be so, which requires that we have generally
0 = (-: — ZjY whence by integrating we get
y =r 1+ m. m +n.m2;
1, m, n, being arbitrary constants.
If we fix the origin of the radii in the middle of the axis of revolution,
and take for a the half of this axis, y will be nothing when m = I and
when m = — 1, which gives m = 0 and 11 = — 1 ; the value of y thus
becomes, 1 (1 — m2); substituting in the equation of equilibrium,
const. = | a « y — ay y' d p d q sin. p — £ g (1 — m *) ;
1 "" X
we shall find a 1 = - — ^ = -r a <p, a <p being the ratio of the centrifugal
16 K 4>
force to the equatorial gravity, a ratio which is very nearly equal to ^ ;
the radius of the spheroid will therefore be
. {1+^(1 -m')};
whence it follows that the spheroid is an ellipsoid of revolution, which is
conformable to what precedes.
BOOK I.] NEWTON'S PRINCIPIA. 269
Thus we have determined directly and independently of series, the
figure of a homogeneous spheroid of revolution, which turns round its
axis, and we have shown that it can only be that of an ellipsoid which
becomes a sphere when <p = 0 ; so that the sphere is the only figure of
revolution which would satisfy the equilibrium of an imrnoveable homo
geneous fluid mass.
Hence we may conclude generally, that if the fluid mass is sollicited
by any very small forces, there is only one possible figure of equilibrium .
or, which comes to the same, there is only one radius a (1 + « y) which
can satisfy the equation of equilibrium,
const. = £ a it . y — a,/y d p . d q' sin. p — N;
y being a function of 6 and of the longitude •&, and y' being what y be
comes when we change Q and » into (i and «•'. Suppose, in fact, that
there are two different rays a (1 + ay) and a(l + «y + «v) which
satisfy this equation ; we shall have
const. = £ a f (y -j- v) — af(y' + v/) d p d q' sin. p — N.
Taking the preceding equation from this, we shall have
const. = £ it v — y v' d p d q sin. p.
This equation is evidently that of a homogeneous spheroid in equili
brium, whose radius is a (1 + a v), and which is not sollicited by any
force extraneous to the attraction of its molecules. The angle -a disappear
ing in this equation, the radius a (1 + a v) will still satisfy it if -a be suc
cessively changed to «• + d -a, -a -f 2 d &, &c., whence it follows, that if
we call v1} v2, &c. what v becomes in virtue of these changes; the
radius
n £1 + avdw-f-avidw + av2dw+ &c.},
or
a (1 + a/v d *r),
will satisfy the preceding equation. If we take the integral fv d -a from
•a = 0 to -o = 2 or, the radius a (1 + aj" \ d •&} becomes that of a sphe
roid of revolution, which, by what precedes, can only be a sphere : see
the condition which results for v.
Suppose that a is the shortest distance of the center of gravity of the
spheroid whose radius is a (I + a v), to the surface, and fix the pole or
origin of the angle 6 at the extremity of a ; v will be nothing at the pole,
and positive every where else; it will be the same for the integraiyVd -a.
But, since the center of gravity of the spheroid whose radius is a (l+«v),
is at the center of the sphere whose radius is a, this point will, in like
manner, be the center of gravity of the spheroid whose radius is
S70 A COMMENTARY ON; [SECT. XII. & XIII.
a (1 -f. ufv d ») ; the different radii drawn from this center to the sur
face of this last spheroid are therefore unequal to one another, if v is not
nothing ; there can only therefore be a sphere in the case of v = 0 ; thus we
learn for a certainty, that a homogeneous spheroid, sollicited by any small
forces whatever, can only be in equilibrium in one manner.
571. We have supposed that N is independent of the figure of
the spheroid; which is what very nearly takes place when the forces,
extraneous to the action of the fluid molecules, are due to the centri
fugal force of rotatory motion, and to the attraction of bodies exterior
to the spheroid. But if we conceive at the center of the spheroid a finite
force depending on the distance r, its action upon the molecules placed at
the surface of the fluid, will depend on the nature of this surface, and
consequently N will depend upon y. This is the case of a homogeneous
fluid mass which covers a sphere of a density different from that of the
fluid ; for we may consider this sphere as of the same density as the fluid,
and .may place at its center a force reciprocal to the square of the dis
tances; so that, if we call c the radius of the sphere, and fits density, that
of the fluid being taken for unity, this force at the distance r will be equal
3 / 1 N
to * K . — — Y - • Multiplying by the element — d r of its direction
c3 fp ])
the integral of the product will be £ <n . — — -, a quantity which we
must add to ae N ; and since at the surface we have r = a (1 + a y), in
the equation of equilibrium of the preceding No., we must add to N,
This equation will become
4 C6 CT j . C I r f i i • XT
const. - —5— "5 1 + (g— 1) . — f y — »/y d p . d q sin. p — N.
-~ o v. a J
If we denote by a (1 + ay + a v), a new expression of the radius of
the spheroid in equilibrium, we shall have to determine v, the equation
f ^ 1
const. — £ T 1 1 + (s — J) r*j — / v/ tl P d ll' sin- P 5
an equation which is that of the equilibrium of the spheroid, supposing it
immoveable, and abstracting every external force.
If the spheroid is of revolution, v will be a function of cos. 6 or m only;
but in this case we may determine it by the analysis of the preceding No. ;
for if we differentiate this equation i + 1 times successively relatively to
in, we shall have
0 = i T -f 1 -f fi-
_
a3' -d m'
BOOK I.] NEWTON'S PR1NCIP1A. 271
but we have
/d p d q' sin. p cos. 2 ' + 2 p = ^ .-^ ;
the preceding equation may therefore be put under this form,
(2i + 3
0=/dpdq'siii.pcos.*' + 8p _ —
We may take i such that, abstraction being made of the sign, we have
-Me -.>
Supposing, therefore, that i is the smallest positive whole number which
renders this quantity greater than unity, we may see, as in the preceding No.,
/d ' + 1 v\
that this equation cannot be satisfied unless we suppose (-= - njri) — °»
which gives
v = mi + Ami-1-i-Bmi-2 + &c.
Substituting in the preceding equation of equilibrium for v, this value,
and for v'
m'5 + A m'1-1 + Bm' i-2 + Sec.
m' being by the preceding No. equal to m cos. 2 p — sin. 2 p cos. q', first
we shall find
which supposes g equal to or less than unity ; thus, whenever a, c, and g
are not such as to satisfy this equation, i being a positive whole number,
the fluid can be in equilibrium only in one manner. Then we shall have
so that
there are, therefore, generally two figures of equilibrium, since a v is sus
ceptible of two values, one of which is given by the supposition of « = 0,
and the other is given by the supposition of v being equal to the preced
ing function of m.
If the spheroid has no rotatory motion, and is not sollicited by any ex
traneous force, the first of these two figures is a sphere, and the second
has for its meridian a curve of the order i. These two curves coincide in
the case of i = J, because the radius a (1 + am) is that of a sphere in
which the origin of the radii is at the distance a from its center , but then
it is easy to see that e = 1, that is, the spheroid is homogeneous, a result
agreeing with that of the preceding No.
272 A COMMENTARY ON [SECT. XII. & XIII.
572. When we have figures of revolution which satisfy the equilibrium,
it is easy to obtain those which are not of revolution by the following
method. Instead of fixing the origin of the angle 6 at the extremity of
the axis of revolution, suppose it at the distance 7 from this extremity, and
call ff the distance from this same extremity of the point of the surface
whose distance from the new origin of the angle 6 is 6. Call, moreover,
ta — /3 the angle comprised between the two arcs 6 and 7 ; we shall
have
cos. (f = cos. 7 cos. 6 + sin. 7 sin. 6 . cos. (w — /3) ;
designating therefore by r . (cos. tf) the function
the radius of the immoveable spheroid in equilibrium, which we have seen
is equal to a {1 + " r- (cos- $')}> wu*l be
a + « a r. {cos. 7 . cos. 6 + sin. 7 . sin. 6 cos. (& — (3)} ;
and although it is a function of the angle *r, it belongs to a solid of revo
lution, in which the angle d is not at the extremity of the axis of revo
lution.
Since this radius satisfies the equation of equilibrium, whatever may be
a, /3, and 7, it will also satisfy in changing these quantities into a', /3', 7',
a"> |8", 7") &c. whence it follows that this equation being linear, the radius
a + a a r . {cos. 7 cos. Q + sin. 7 sin. Q cos. (« — /3 )}
+ a' aT . [cos. y cos. 6 -f- sin. y* sin. 6 cos. (^ — j3')]
+ &c.
will likewise satisfy it. The spheroid to which this radius belongs is no
longer one of revolution ; it is formed of a sphere of the radius a, and of
any number of shells similar to the excess of the spheroid of revolution
whose radius is a + a a r . (m) above the sphere whose radius is a, these
shells being placed arbitrarily one over another.
If we compare the expression of r. (cos. $') with that of P (i> of No. 567,
we shall see that these two functions are similar, and that they differ only
by the quantities 7 and /3, which in P W are v ai}d ^ and by a factor in
dependent of m and vr ; we have, therefore,
d
It is easy hence to conclude, that if we represent by a Y (i> the function
a . r . {cos. 7 cos. 6 -j- sin. 7 sin. d . cos. (-a — (3 )}
+ a! . r . {cos. 7' cos. Q + sin. 7' sin. 6 . cos. (v — /3')}
BOOK I.] NEWTON'S PRINCIP1A. 273
Y (l) will be a rational and entire function of m, VI — nT2 cos. •&,
VI — m 2 sin. *, which will satisfy the equation of partial differences,
•
choosing for Y ®9 therefore, the most general function of that nature, the
function a (1 + « Y (i)) will be the most general expression of the equili
brium of an immoveable spheroid.
We may arrive at the same result by means of the series for V in 555 ;
for the equation of equilibrium being, by the preceding No.,
const. = V + a2 N;
if we suppose that all the forces extraneous to the reciprocal action of the fluid
molecules, are reducible to a single attractive force equal to f it. • C ,
placed at the center of the spheroid, by multiplying this force by the ele
ment — d r of its direction, and then integrating, we shall have
and since at the surface r = a (1 + « y) the preceding equation of equi
librium will become
c 3
const. = V + t « » . — (1 — f)y.
fl
Substituting in this equation for V its value given by formula (3) of
No. 555, in which we shall put for r its value a (1 -f a y), and by sub
stituting for y its value
Y<°> + YW + Y® + &c.;
we shall have
°=
the constant a being supposed such, that const. = $ ir a 2. This equation
gives Y ») = 0, Y ^ = 0, Y ® = 0, &c. unless the coefficient of one of these
quantities, of Y W for example, is nothing, which gives
(I x c3 __ 2 i — 2
~s'~tir~~ 2i + 1 '
i being a positive whole number, and in this case all these quantities ex
cept Y W are nothing ; we shall therefore have y = Y (i>, which agrees
with what is found above.
Thus we see, that the results obtained by the expansion of V into a se-
VOL. II. S
274 A COMMENTARY ON [SECT. XII. & XIII.
ries, have all possible generality, and that no figure of equilibrium has
escaped the analysis founded upon this expansion ; which confirms what
we have seen a priori, by the analysis of 555, in which we have proved
that the form which we have given to the radius of spheroids, is not arbi
trary but depends upon the nature itself of their attractions.
573. Let us now resume equation (J) of No. 567. If we therein sub-
stitutefor V its value given by formula (6) of No. 558, we shall have rela
tively to the different fluid shells
/{] TT f
•Hsfcr/f ft*4 4««r/f d
n r
a W+y
+ a r2 JZ<°> + Z® + r Z® + r2 Z^ + &c.} ; . . . . (1)
the differentials and integrals being relative to the variable a; the two first
integrals of the second member of this equation must be taken from a = a to
a = 1, a being the value of a, relative to the leveled fluid shell, which we are
considering, and this value at the surface being taken for unity : the two last
integrals ought to be taken from a = 0 to a = a : finally, the radius r
ought to be changed into a ( 1 + ay) after all the differentiations and in
tegrations. In the terms multiplied by a it will suffice to change r into
a ; but in the term -^- f % d . a3 we must substitute a (1 + a y) for r ;
o r
which changes it into this
4 ?r
3 a *.-•'.•'
and consequently, into the following
_ « Y(1> — a Y(3) — &C.L fp d a3.
w «.*
Hence if in equation (1) we compare like functions, we shall have
AC-— = 2 * fe d a2 + 4 a f fp d (a2 Y^x -1-— — r~ A -3
J 0 J *> t/ & V
3 a
«> ;t a
the two first integrals of the second member of this equation being taken
from a = a to a = 1, the three other integrals must be taken from a
= 0 to a = a. This equation determining neither a nor Y (0), but only a
relation between them, we see that the value of Y (0) is arbitrary, and may
be determined at pleasure. We shall have then, i being equal to, 01
greater than unity,
BOOK I.] NEWTON'S PRINCIPIA. £73
4 » a ' ^ j /Y Ci)\ 4 cr „ , ,, , ,
0 = — : fe d. ( — r-3 j — = — Y ]) ft d a3
2i+lJ s Va'-'V 3 a J £
the first integral being taken from a — a, to a = ], and the two others
being taken from a = 0 to a = a. This equation will give the value of
Y (i) relative to each fluid shell, when the law of the densities g shall be
known.
To reduce these different integrals within the same limits, let
4 T
the integral being taken from a = 0 to a = 1 ; Z' (i) will be a quantity in
dependent of a, and the equation (2) will become
— 3/g d (a * + s Y «) — 3 a 2 5 + 1 Z' (i> ;
all the integrals being taken from a = 0 to a = a.
We may make the signs of integration disappear by differentiating re
latively to a, and we shall have the differential equation of the second
order,
/d'Y«x Ji(j+ 1) 6 g a 1 6ga2 /d Y ®*
\daz)' \ a2 /fda3/ /g d. a3 V d a )'
The integral of this equation will give the value of Y (l) with two arbi
trary constants ; these constants are rational and entire functions of the
order i, of m, VI — m 2 . sin. •&, and VI — m 2 . cos. ^-, such, that re
presenting them by U (i), they satisfy the equation of partial differences,
dm / 1— m2
One of these functions will be determined by means of the function
Z' (i) which disappears by differentiation, and it is evident that it will be a
multiple of this function. As to the other function, if we suppose that
the fluid covers a solid nucleus, it will be determined by means of the
equation of the surface of the nucleus, by observing that the value of
Y :i) relative to the fluid shell contiguous to this surface, is the same as
that of the surface. Thus the figure of the spheroid depends upon the
figure of the internal nucleus, and upon the forces which sollicit the
fluid.
574. If the mass is cntirebjiKfluid, nothing then determining one of the
arbitrary constants, it would seem that there ought to be an infinity of
S 2
276 A COMMENTARY ON SECT. XII. & XIII.
figures of equilibrium. Let us examine this case particularly, which is
the more interesting inasmuch as it appears to have subsisted primi
tively for the celestial bodies.
First, we shall observe that the shells of the spheroid ought to decrease
in density from the center to the surface ; for it is clear that if a denser
shell were placed above a shell of less density, its molecules would pene
trate into the other in the same manner that a ponderous body sinks into
a fluid of less density ; the spheroid will not therefore be in equilibrium.
But whatever may be its density at the center, it can only be finite ; re
ducing therefore the expression of g into a series ascending relatively to
the powers of a, this series will be of the form /3 — y . a n — &c. |8, y and
n being positive ; we shall thus have
3 y . a " & .
(n + 3) /3
and the differential equation in Y w will become
To integrate this equation, suppose that Y (i) is developed into a series
ascending according to the powers of a, of this form
YU) = as. U« + a5'. U' « + &c. ;
the preceding differential equation will give
' — i + 2) as'~2 U'<» + &c.
= j_ (g+ 1)a.-S.U0>+(8>+i)a*-au'»+&c.S . (e)
(n + A) P
Comparing like powers of a, we have (s + i + 3) (s — i + 2) = 0,
which gives « = i — 2, and s = — i — 3. To each of these values of
s, belongs a particular series, which, being multiplied by an arbitrary, will
be an integral of the differential equation in Y (i> ; the sum of these two in
tegrals will be its complete integral. In the present case, the series which
answers to s = — i — 3 must be rejected ; for there thence results for a
Y (i) , an infinite value, when a shall be infinitely small, which would render
infinite the radii of the shells which are infinitely near to the center. Thus
of the two particular integrals of the expression of Y w , that which answers
to s = i — 2 ought alone to be admitted. This expression then, contains
no more than one arbitrary which will be determined by the function Z (i) .
Z (1> being nothing by No. 567, Y (1) is likewise nothing, so that the
center of gravity of each shell, is at the center of gravity of the entire
BOOK I.] NEWTON'^ PRINC1PIA. 277
spheroid. In fact the differential equation in Y (i) of the preceding No.
gives
/d'YWv . /2x 6g
Vda2/"VaV /gd.
a)_ 6ga
3* /gd.
We satisfy this equation by making Y (1) = - , U (l) being indepen-
ft
dent of a. This value of Y(1) is that which answers to the equation
s = { — 2 ; it is, consequently, the only one which we ought to admit.
Substituting it in the equation (2) of the preceding No., and supposing
Z (1) = 0, the function U (1) disappears, and consequently remains arbitrary;
but the condition that the origin of the radius r is at the center of gravity
of the terrestrial spheroid, renders it nothing ; for we shall see in the follow
ing No. that then Y (1) is nothing at the surface of every spheroid covered
over with a shell of fluid in equilibrium ; we shall have, therefore, in the
present ease U (1) = 0 ; thus, Y (1) is nothing relatively to all the fluid shells
which form the spheroid.
Now consider the general equation,
Y« = as . U« + as/ . U' « + &c. ;
s being, as we have seen, equal to i — 2, s is nothing or positive, when i
is equal to or greater than 2; moreover, the functions U' w, U//(l), &c. are
given in U (i), by the equation (e) of this No. ; so that we have
Y« = h. U(i>;
h being a function of a, and U ('° being independent of it. If we substi
tute this value of Y ® in the differential equation in Y l'°, we shall have
d2h f 6g a3 ) _h_ 6 ga2 dji
"cl"a^"=: V(l ' 1)""77d7a"= rj • a2 "" /gd.a3 'da'
The product i (i + 1) is greater than -7*— 4 - r> when i is equal to or
t/ fa
e a3
greater than 2, for the fraction ,. g ' - - is less than unity : in fact its
J S d • a
denominator f § d . a 3 is equal to § a 3 — f a 3 d g, and the quantity
— fa3 d g is positive, since g decreases from the center to the surface.
Hence it follows that h and -r— - are constantly positive, from the
center to the surface. To show this, suppose that both these quantities are
positive in going from the center; d h ought to become negative before h,
and it is clear that in order to do this it must pass through zero ; but
from the instant it is nothing, d2 h becomes positive in virtue of the pre
ceding equation, and consequently d h begins to increase ; it can never
therefore become negative. Whence it follows that h and d h always pre-
S3
278 A COMMENTARY ON [SECT. XII. & XIII.
serve the same sign from the center to the surface. Now both of these
quantities are positive in going from the center ; for we have in virtue of
equation (e), s' — 2=s + n — 2, which gives s' = i + n — 2 ; hence
we have
s' + i + 3) (S'- i + a) u- B = 6 n 5 +
Ufl>
Vn + 3) p
whence we derive
U(i)_ 6(i-l)y.UM
(n + 8)(2i + n + l)./3'
we shall therefore get
6 (i— 1) y. a1*"-2
h = al-3+(n + 3)(2i + n
dh 6i-i
i SU'-8 . - -.
d a - 1 (n+3)(2i + n+l)/3 ,+
7, /?, n, being positive, we see that at the center h and d h are positive,
when i is equal to or greater than 2 ; they are therefore constantly positive
from the center to the surface.
Relatively to the Earth, to the Moon, to Jupiter, &c. Z (i> is nothing or
insensible, when i is equal to or greater than 3 ; the equation (2) of the
preceding No. then becomes
0= ^3a^
the first integral being taken from a = a, to a = 1, and the two others
being taken from a = 0, to a = a. At the surface where a = 1, this equa
tion becomes
0 = { — (2 i+ 1) h/£d. a3 + 3/£d (a^h)}. U«;
an equation which we can put under this form
0 = J_(2i — 2)gh + (2i+l) h/a3dg— 3/a i + Mi.d^ U «.
d g is negative from the center to the surface, and h increases in the
same interval; the function (2 i + 1) \\f a 3 d g — 3y"a ' + 3 h d g is therefore
negative in the same interval ; thus in the preceding equation the coeffi
cient of U (i) is negative and cannot be nothing at the surface ; U (i) ought
therefore to be nothing, which gives Y w = 0 ; the expression of the ra
dius of the spheroid thus reduces to a + a a {Y (0) + Y (2)] ; that is to say,
that the surface of each leveled shell of the spheroid is elliptic, and conse
quently its exterior surface is elliptic.
Z (2>, relatively to the Earth is, by No. 567, equal to — £- (m 2 — £ ) ;
& €X§
the equation (2) of the preceding No. gives therefore
BOOK I.] NEWTON'S PR1NCIPIA. 279
0= &«r
At the surface, the first integral^ d h is nothing; we have therefore at
this surface where a = 1,
|J(8) =
Let a p, be the ratio of the centrifugal force to the equatorial gravity ;
the expression of gravity to quantities of the order a, being equal to
I ""ft d . a3; we shall have g = f ir a ipfg d . a3; wherefore
U(2> =
2 /g.d(a*b);
2h— =•
5 j g . a ' a a
comprising therefore in the arbitrary constant a, what we have taken for
unity, the function
a h <p
5 * f i .a2 d a
the radius of the terrestrial spheroid at the surface will be
a h p (1 — m2)
•*•! O /• _ ,1 / „ 5 K \ *
5 ' y"g.a2 d a
The figure of the earth supposed fluid, can therefore only be that of an
ellipsoid of revolution ; all of whose shells of constant density are elliptic,
and of revolution, and in which, the ellipticities increase, and the densities
decrease from the center to the surface. The relation between the ellip
ticities and densities is given by the differential equation of the second
order,
d'h _ 6_h / ga3 \ 2ga2 dji
da2 " a2 X. ~ 3/£ a 2 d a/ ~fg . a " d a ' cTa '
This equation is not integrable by known methods except in some par
ticular suppositions of the densities g ; but if the law of the ellipticities
were given, we should easily obtain that of the corresponding densities.
We have seen that the expression of h given by the integral of this equa
tion contains, in the present question, only one arbitrary, which disappears
from the preceding value of the radius of the spheroid ; there is therefore
only one figure of equilibrium differing but little from a sphere, which is
possible, and it is easy to see that the limits of the flattening of this figure
are ^ and £ « <p, the former of which corresponds to the case where all
280 A COMMENTARY ON [SECT. XII. &XI11.
the mass of the spheroid is collected at its center, and the second to the
case where this mass is homogeneous.
The directions of gravity from any point of the surface to the center do
not form a straight line, but a curve whose elements are perpendicular to
the leveled shells which they traverse : this curve is the orthogonal tra
jectory of all the ellipses which by their revolution form these shells. To
determine its nature, take for the axis, the radius drawn from the center
to a point of the surface, d being the angle which this radius forms with
the axis of revolution. We have just seen that the general expression of
any shell of the spheroid is a + a k . a h . (1 — m2), k being independent
of a : whence it is easy to conclude that if we call a y', the ordinate let
fall from any point of the curve upon its axis, we shall have
/ i „ ( rh d a)
ay = a a k . sin. 2 6 -\ c — /— Y ,
c being the entire value of the integral /* - , taken from the center to
the surface.
575. Now consider the general case in which the spheroid always fluid
at its surface, may contain a solid nucleus of any figure whatever, but dif
fering but little from the sphere. The radius drawn from the center of
gravity of the spheroid to its surface, and the law of gravity at this sur
face have some general properties, which it is the more essential to con
sider, inasmuch as these properties are independent of every hypothesis.
The first of these properties is, that in the state of equilibrium the
fluid part of the spheroid must always be disposed so, that the function
Y (1) may disappear from the expression of the radius drawn from the cen
ter of gravity of the whole spheroid to its surface ; so that the center of
gravity of this surface coincides with that of the spheroid.
To show this, we shall observe that R being supposed to represent the
radius drawn from the center of gravity of the spheroid to any one of its
molecules, the expression of this molecule will be g R 2. d R . d m . d »,
and we shall have by 556, in virtue of the properties of the center of
gravity,
0 =/g R3. dR.dm.d^.m;
0 = R 3. d R . d m . d w . V 1 — m2. sin. w;
0 =/g R 3. d R.dm.dw. V 1 — m ". cos. ».
Conceive the integral f g R 3. d R taken relatively to R from the origin
of R to the surface of the spheroid, and then developed into a series of
the form
&c. ;
BOOK I.] NEWTON'S PRINCIPIA.
N (i) being whatever i may be, subject to the equation of partial differ
ences
d m
we shall have by No. 556, when i is different from unity,
0 = /N«. m d m . d * ; 0 =/N w. d m . d » . V I — in 2. sin.
and
0=/Nu>.dm.dw. VI — m2. cos. *.
The three preceding equations given by the nature of the center of
gravity, will become
0 =/N<1>mdm.d«j 0 =/N^dm.dw. V 1— m2.sin.*r;
0 =/N (1> d m . d * . V 1 — m2. cos. 9.
N U) is of the form
H m + H'. V I — m2. sin. * + H". V 1 — m 2. cos. «.
Substituting this value, in these three equations, we shall have
H = 0; H7 = 0; H" = 0;
where N (1) = 0 ; this is the condition necessary that the origin of II is at
the center of gravity of the spheroid.
Now let us see, what N (1) becomes relatively to the spheroids differing
little from the sphere, and covered over with a fluid in equilibrium. In
this case we have R = a (1 + a y), and the integral fg. R 3. d R, be
comes £./£ d . [a* (1 + 4 a y)}, the differential and integral being rela
tive to the variable a, of which g is a function. Substituting for y its va
lue Y<°> + Y « + Y® + &c., we shall have
N<» = a/gd (a4 Y»>).
The equation (2) of No. 573 gives, at the surface where a = 1, and
observing that Z U) is nothing
/fd(a*Y<») = Y'Vfd.a',
the value of Y(1) in the second member of this equation, being relative to
the surface ; thus, N (1) being nothing, when the origin of R is at the cen
ter of gravity of the spheroid, we have in like manner Y ^ =r 0.
576. The permanent state of equilibrium of the celestial bodies, makes
known also some properties of their radii. If the planets did not turn ex
actly, or at least if they turned not nearly, round one of their three principal
axes of rotation, there would result in the position of their axes of rota
tion, changes which for the earth above all would be sensible ; and since
the most exact observations have not led to the discovery of any, we may
conclude that long since, all the parts of the celestial bodies, and princi-
282 A COMMENTARY ON [SECT. XII. & XIII.
pally the fluid parts of their surfaces, are so disposed as to render stable
their state of equilibrium, and consequently their axes of rotation. It is
in fact very natural to suppose that after a great number of oscillations,
they must settle in this state, in virtue of the resistances which they suffer.
Let us see, however, the conditions which thence result in the expression
of the radii of the celestial bodies.
If we name x, y, z the rectangular coordinates of a molecule d M of
the spheroid, referred to three principal axes, the axis of x being the axis
of rotation of the spheroid ; by the properties of these axes as shown in
dynamics, we have
0=/xy.dM; 0=/xz.dM; 0=/yz.dM;
the integrals ought to be extended to the entire mass of the spheroid,
R being the radius drawn from the origin of coordinates to the molecule
d M ; 6 being the angle formed by R and by the axis of rotation ; and
•a being the angle which the plane formed by this axis and by R, makes
with the plane formed by this axis and by that of the principal axes, which
is the axis of y ; we shall have
x=Rm; y = R V 1 — mz. cos. ^ ; z = R V 1 — m2. sin. -a ;
dM = gR*d Rdm.d^.
The three equations given by the nature of the principal axes of rota
tion, will thus become
0 = J 'g .R4. dR.dm.dar.m V 1 — m2. cos. -a ;
0 =fs . R4. dR.dm.d^.m VI— m2. sin. »;
0 =/g.R4. d R.dm.d ».(! — m2) sin. 2*.
Conceive the integral fg R4 d R taken relatively to R, from R = 0,
to the value of R at the surface of the spheroid, and developed into a
series of the form U (0> + U (1) + U (2) + &c. ; U (i) being, whatever i may
be, subject to the equation of partial differences,
m. — ,
We shall have by the theorem of No. 556, where i is different from 2,
and by observing that the functions m V 1 — m z. cos. •#, m V 1 — m 2. sin. «,
and (1 — m2) sin. 2 v, are comprised in the form Uff- ;
0 = / U (l). d m . d -a . m . VI — mz. cos. «r ;
0 = /U W. d m . d « . m . V 1 — m 2. sin. » ;
0 =/U (i>. dm.d ».(! — m 2) sin. 2 ».
BOOK I.] NEWTON'S PRINCIPIA. 283
The three equations relative to the nature of the axes of rotation, will
thus become
0 =r / U (8>. d m . d * . m . VI — m 2. cos. v ;
0 = /U W. d m . d «. m . V 1 — m 2. sin. * ;
0 = /U®. dm.dw. (1 — m2) sin. 2 w.
These equations therefore depend only on the value of U (2) : this value
is of the form
H (m 2 — I) + H' m V 1 — m 2. sin. «• + H" m V 1 — m2. cos. » +
H'" (1 — m2) sin. 2 * + H"" (1 — m2) cos. 2 »:
substituting it in the three preceding equations, we shall have
H' = 0; H" = 0; H'" = 0.
It is to these three conditions that the conditions necessary to make the
three axes of x, y, z the true axes of rotation are reduced, and then U (2)
will be of the form
H (m2 — i) + H"" (1 — m2) cos. 2 ~.
When the spheroid is a solid differing but little from the sphere, and
covered with a fluid in equilibrium, we have R = a (1 + « y), and con
sequently
ft R4. d R = £/g d. {a5. (1+ 5 a y)}.
If we substitute for y, its value Y <°> + Y (1) + Y (~> + &c. ; we shall
have
U® = a/^d (a5 Y(2>).
The equation (2) of No. 573, gives for the surface of the spheroid,
~fs d (a5 Y«) = | * YW/f d- a3 — Z^2';
Y w and Z (2) in the second member of this equation being relative to the
surface ; we have therefore,
U® = f a YCygd.a3 — 5aZ(8).
The value of Z'2) is of the form
— -f- (m 2 — i) + g' m V 1 — m2. sin. * + g"
+ g"'(l— ms)sin.2w + gr///(l — m2) cos.
and that of Y® is of the form
m Vn2. Cos.
— h (m 2 — i) + h' m V 1— in2, sin. * + h" m V 1 — m 2. cos. «•
+ li/x/. ( 1 — m 2) sin. 2 * + hw/ (1 — m 2) cos. 2 *r.
Substituting in the preceding equation, these values, and H (m 2 _
+ H"" (1 — m 2) cos 2 w, for U ® ; we shall have
284 A COMMENTARY ON [SECT. XII. & XIII.
Such are the conditions which result from the supposition that the sphe
roid turns round one of its principal axes of rotation. This supposition
determines the constants h', h", h"' by means of the values g', g", g"' ;
but it leaves indeterminate the quantities h and hx/// as also the functions
Y<3>, Y<«, &c.
If the forces extraneous to the attraction of the molecules of the sphe
roid are reduced to the centrifugal force due to its rotatory motion ; we
shall have g' = 0, g" = 0, g" = 0 ; wherefore h' = 0, h" = 0, h"' = 0,
and the expression of Y l'2), will be of the form
— h (m 2 — J) + h"" (1 — m 2) cos. 2 v.
577. Let us consider the expression of gravity at the surface of the
spheroid. Call p this force ; it is easy to see by No. 569, that we shall
have its value by differentiating the second member of the equation (1) of
573 relatively to r, and by dividing its differential by — d r ; which gives
at the surface
— «r [2Z®> + 2Z® + 3r.Z® + 4r*. Z<4> + &c.} ;
these integrals being taken from a = 0, to a = 1. The radius r at the
surface is equal to 1 -f a y, or equal to
1 + «£Y(0) + YW 4. Y® + &c.};
we shall hence obtain
P = ~---~
+ 4«<r/gd. {a3Y<°> + ^Y«) + ?^-Y (2)+ &c.}
o O
— a {2 Z «» + 2 Z fa> + 3 Z ® + 4 Z W + &c. j.
The integrals of this expression may be made to disappear by means of
equation (2) of No. 573, which becomes at the surface,
a1*3 Y « = *Y «.*. a'- Z'O.
supposing therefore
P=f<r/gd.a3 —
o
we shall have
p = P + aP. {Y<8> +
— a £5 Z ® + 7 Z <3> + 9 Z w + . . . + (2 i+ 1) Z (i> + &c.}.
By observations of the lengths of the seconds' pendulum, has been re
cognised the variation of gravity at the surface of the earth. By dy
namics it appears that these lengths are proportional to gravity ; let
BOOK L] NEWTON'S PRINCIPIA. 285
therefore 1, L be the lengths of the pendulum corresponding to the gravi
ties p, P ; the preceding equation will give
Relatively to the earth a Z(2) reduces by 567, to — -^ (m 2 — i), or,
0
which comes to the same, to -- -^. P. (m2 — ^), a <p being the ratio of
0
the centrifugal force to the equatorial gravity; moreover, Z(3), Zw, &c.
are nothing ; we have therefore
1 = L + a L. JYW + 2 Y(3> + 3 Y<4> + . . . + (i — 1) Y«J
The radius of curvature of the meridian of a spheroid which has for its
radius 1 + a y, is
l + .(lHLZ) + ._ - -.n
\ d m / \ dm /
designating therefore by c, the magnitude of the degree of a circle whose
radius is what we have taken for unity ; the expression of the degree of
'
raus s wat we ave taen o
the spheroid's meridian, will be
—
dm
y is equal to Y^ + Y^ + Y(2> + &c. We may cause Y^ to disap
pear, by comprising it in the arbitrary constant which we have taken for
the unit ; and Y <l> by fixing the origin of the radius at the center of gravity
of the entire spheroid. This radius thus becomes,
1 + a jy<2> + Y<3> + Y<4> + &c.}.
If we then observe that
the expression of the degree of the meridian will become
c —
f /d Y^x , /d Y<3\ , \
acm<(- -) + ( -3 --- ) + &C. f
(_ \ d m / v d m /
a c. ;
1 — m
286 A COMMENTARY ON [SECT. XII. & XIII.
If we compare these expressions of the terrestrial radius with the length
of the pendulum, and the magnitude of the degree of the meridian, we
see that the term a Y (i) of the expression of the radius is multiplied by
i — 1, in the expression of the length of the pendulum, and by i 2 + i — 1
in that of the degree ; whence it follows, that whilst i — 1 is considerable,
this term will be more sensible in the observations of the length of the
pendulum than in that of the horizontal parallax of the moon which is
proportional to the terrestrial radius ; it will be still more sensible in the
measures of degrees than in the lengths of the pendulum. The reason of
it is, that the terms of the expression of the terrestrial radius undergo two
variations in the expression of the degree of the meridian ; and each dif
ferentiation multiplies these terms by the corresponding exponent of m,
and this renders them the more considerable. In the expression of the
variation of two consecutive degrees of the meridian, the terms of the ter
restrial radius undergo three consecutive differentiations; those which
disturb the figure of the earth from that of an ellipsoid, may thence be
come very sensible, and the ellipticity obtained by this variation may be
very different from that which the observed lengths of the pendulum give.
These three expressions have the advantage of being independent of the
interior constitution of the earth, that is to say, of the figure and density
of its shells; so that if we are going to determine the functions Y(2), Y(3),
&c. by measures of degrees of meridians and parallaxes, we shall have
immediately the length of the pendulum ; we may therefore thus ascertain
whether the law of universal gravity accords with the figure of the earth,
and with the observed variations of gravity at its surface. These remark
able relations between the expressions of the degrees of the meridian and
of the lengths of the pendulum may also serve to verify the hypotheses
proper to represent the measures of degrees of this meridian : this will be
perceptible from the application we now proceed to make to the hypothe
sis proposed by Bouguer, to represent the degrees measured northward
in France and at the equator.
Suppose that the expression of the terrestrial radius is 1 + « Y (2) +
a. Y (4), and that we have
= _ B m4 _ m« +
it is easy to see that these functions of in satisfy the equations of partial
differences which Y(2) and Y(4> ought to satisfy. The variation of the de
grees of the meridian will be, by what precedes,
{3 A — ~2B} 2+ 15acB.m4.
a c
BOOK I.] NEWTON'S PRINCIPIA. 287
Bouguer supposes this variation proportional to the fourth power of the
sine of the latitude, or, which nearly comes to the same, to m4; the term
multiplied by m 2, therefore, being made to disappear from the preceding
function, we shall have
B = !.A;
thus in this case the radius drawn from the center of gravity of the earth
at its surface, will be in taking that of the equator for unity,
7 a A .
1 — 3T~ (4m +m4).
The expression of the length 1 of the pendulum, will become, denoting
by L, its value at the equator,
L + f « f . L m2 — %^L (16 m2 + 21 m4).
o4i
Finally, the expression of the degree of the meridian, will be, calling c
its length at the equator,
105
c + — . « A . c . m 4.
We shall observe here, that agreeably to what we have just said, the
term multiplied by m 4 is three times more sensible in the expression of
the length of the pendulum than in that of the terrestrial radius, and five
times more sensible in the expression of the length of a degree, than in
that of the length of the pendulum ; finally, upon the mean parallel it
would be four times more sensible in the expression of the variation of
consecutive degrees, than in that of the same degree. According to Bou-
959
guer, the difference of the degrees at the pole and equator is ; it is
Ot) i Do
the ratio which, on his hypothesis, the measures of degrees at Pello, Paris
105
and the equator, require. This ratio is equal to -=-.- . a A ; we have
<34
therefore
a A = 0. 0054717.
Taking for unity the length of the pendulum at the equator, the va
riation of this length, in any place whatever, will be
0. 0054717
-. [IG m s + 21 m4} + f a <p . m «.
By No. 563, we have a p = 0. 00345113, which gives f a p = 0. 0086278,
and the preceding formula becomes
0. 0060529. m 2 — 0. 0033796. m 4.
288 A COMMENTARY ON [SECT. XII. & XIII.
At Pello, where m = sin. 74°. 22', this formula gives 0.0027016 for
the variation of the length of the pendulum. According to the observa
tions, this variation is 0.0044625, and consequently much greater; thus,
since the hypothesis of Bouguer cannot be reconciled with the observations
made on the length of the pendulum, it is inadmissible.
578. Let us apply the general results which we have just found, to the
case where the spheroid is not sollicited by any extraneous forces, and
where it is composed of elliptic shells, whose center is at the center of
gravity of the spheroid. We have seen that this case is that of the earth
supposed to be originally fluid : it is also that of the earth in the hypo
thesis where the figures of the shells are similar. In fact, the equation
(2) of No. 573 becomes at the surface where a — 1,
The shells being supposed similar, the value of Y (i) is, for each of
them, the same as at the surface ; it is consequently independent of a, and
we have
When i is equal to or greater than 3, Z ;i) is nothing relatively to the
i + 3
earth; besides the factor 1 — • . . a ' is always positive ; therefore Y ^
is then nothing. Y (1) is also nothing by No. 575, when we fix the origin
of the radii at the center of gravity of the spheroid. Finally, by No. 577,
we have Z (2) equal to
a'da ;
we have therefore
fg&*d a (1 —a2)
Thus the earth is then an ellipsoid of revolution. Let us consider there
fore generally the case where the figure of the earth is elliptic and of re
volution.
In this case, by fixing the origin of terrestrial radii at the center of
gravity of the earth, we have
Y(D - 0; Y<3> = 0; Y« = 0 ; &c.
BOOK I.] NEWTON'S PR1NCIPIA. 289
h being a function of a ; moreover we have
Z(.) _ 0. Z (3) = 0; Z« = 0; &C.
the equation (2) of No. 573 will therefore give at the surface
0 = 6./fd(ash) + 5. (p-2h)/ed.a3 . . . (1)
This equation contains the law which ought to exist to sustain the
equilibrium between the densities of the shells of the spheroid and their
ellipticities ; for the radius of a shell being a [I +a Y (0) — a h (p 2 — |)} 5
if we suppose, as we may, that Y (0) = — ^ h, this radius becomes
a (1 — a h . ,u, 2), and a h is the ellipticity of the shell.
At the surface, the radius is 1 — a h . ^ 2 ; whence we see that the de
crements of the radii, from the equator to the poles, are proportional to
/* 2, and consequently to the square of the sines of the latitude.
The increment of the degrees of the meridian from the equator to the
poles is, by the preceding No., equal to 3 a h c . ^ 2, c being the degree
of the equator ; it is therefore also proportional to the square of the sine
of the latitude.
The equation (1) shows us that the densities being supposed to decrease
from the center to the surface, the ellipticity of the spheroid is less than
in the case of homogeneity, at least whilst the ellipticities do not increase
from the surface to the center in a greater ratio than the inverse ratio of
the square of the distances to this center. In fact, if we suppose h = — 2 ,
we shall have
If the ellipticities increase in a less ratio than — ^ , u increases from tlte
center to the surface, and consequently d u is positive ; besides, d g is ne
gative by the supposition that the densities decrease from the center to the
surface; thus5</( d uya3 d g) is a negative quantity, and making at the
surface
/fd(aMi) = (h-f)/gd.a3,
f will be a positive quantity. Hence equation (1) will give
5 <f> — 6 f
~JT
a h will therefore be less than - - , and consequently it will be less than
VOL. II. T
2<)0 A COMMENTARY ON [SECT. XII. & XIII.
in the case of homogeneity, where d g being equal to nothing f is also equal
to zero.
Hence it follows, that in the most probable hypotheses, the flattening oi
the spheroid is less than — y-r ; for it is natural to suppose that the shells
~r
of the spheroid are denser towards the center, and that the ellipticities
increase from the surface to the center in a less ratio than — -z , this ratio
a
giving an infinite radius for shells infinitely near to the center, which is
absurd. These suppositions are the more probable, inasmuch as they
become necessary in the case where the fluid is originally fluid ; then the
denser shells are, as we have seen, the nearer to the center, and the ellip
ticities so far from increasing from the surface to the center, on the con
trary, decrease.
If we suppose that the spheroid is an ellipsoid of revolution, covered
with a homogeneous fluid mass of any depth whatever, by calling a' the
semi-minor axis of the solid ellipsoid, and a h' its ellipticity, we shall have
at the surface of the fluid,
ft d (a5 h) = h — a" h' +fe d (a5 h)j
the integral of the second member of this equation being taken relatively
to the interior ellipsoid, from its center to its surface, and the density of
the fluid which covers it being taken for unity. The equation (1) will
give for the expression of the ellipticity « h, of the terrestrial spheroid,
_ 5ap jl — a/3+/gda3j — Gah'. a/5 + 6a/gd(a5h) .
4—10 a'3 + 10./gd.a3
the integrals being taken from a = 0 to a
Let us now consider the law of gravity, or which comes to the same,
that of the length of the pendulum at the elliptic surface in equilibrium.
The value of 1, found in the preceding No., becomes in this case
1 = L + « L J| <f> — hj (m8 — £) ;
making, therefore, L' = L — i a L (f p — h), we shall have, in neglecting
quantities of the order a \
1 = L' + «L' (|f — h)//,2;
an equation from which it results that L' is the length of the seconds'
pendulum at the equator, and that this length increases from the equator
to the poles, proportionally to the square of the sine of the latitude.
If we call a t the excess of the length of the pendulum at the pole above
its length at the equator, divided by the latter, we shall have
a t — a (f <p — h);
BOOK I.] NEWTON'S PRINCIPIA. 291
and consequently
ae + ah = ->ap;
a remarkable equation between the ellipticity of the earth and the varia
tion of the length of the pendulum from the equator to the poles. In the
case of homogeneity ah = f a <p ; hence in this case a s = ah; but if
the spheroid is heterogeneous, as much as a h is above or below ^ a <p} so
much is a s above or below the same quantity.
579. The planets being supposed covered with a fluid in equilibrium, it
is necessary, in the estimate of their attractions, to know the attraction of
spheroids whose surface is fluid and in equilibrium : we may express it
very simply in this way. Resume the equation (5) of No. 558 ; the signs
of integration may be made to disappear by means of equation (2) of No.
573, which gives at the surface of the spheroid,
thus fixing the origin of the radii r at the center of gravity of the spheroid
which makes Y (^disappear; then observing that Z (1) is nothing, and that Y (0)
being arbitrary, we may suppose «-. Y^ — Z(0) = 0, the equation (5)
9
of 558, will give
an expression in which we ought to observe that — f% d . a 3 expresses the
o
mass of the spheroid, since, in the case of r being infinite, the value of V
is equal to the mass of the spheroid divided by r. Hence the attraction
of the spheroid parallel to r will be — (-r-) 5 the attraction perpendicu
lar to this radius, in the plane of the meridian will be --
T— \ ; finally, the attraction perpendicular to this same radius in the
direction of the parallel will be
r V 1 -m2
The expression of V, relatively to the earth supposed elliptic, becomes
M being the mass of the earth.
T2
292 A COMMENTARY ON [SECT. XII. & XIII.
580. Although the law of attraction in the inverse ratio of the square
of the distance is the only one that interests us, yet equation (1) of 554
affords a determination so simple of the gravity at the surface of homoge
neous spheroids in equilibrium, whatever is the exponent of the power of
the distance to which the attraction is proportional, that we cannot here
omit it. The attraction being as any power n of the distance, if we de
note by d m a molecule of the spheroid, and by f its distance from the
point attracted, the action of d m upon this point multiplied by the element
— d f of its direction, will be — d ^ f n. d f. The integral of this quantity,
d ^ f n + i
taken relatively to f, is -- , and the sum of these integrals ex
tended to the entire spheroid is -- - — - ; supposing, as in 554, that V =
/f n + l d ft.
If the spheroid be fluid, homogeneous, and endowed with rotatory mo
tion, and not sollicited by any extraneous force, we shall have at the sur
face, in the case of equilibrium, by No. 567,
const. = — jJLj + $ g r» (1 — m *),
r being the radius drawn from the center of gravity of the spheroid at its
surface, and g the centrifugal -force at the distance 1 from the axis of ro
tation.
The gravity p at the surface of the spheroid is equal to the differential
of the second member of this equation taken relatively to r, and divided
by — d r, which gives
1 /d Vx
P = — ; — f • Lr3-*J — ff r (1 — ms).
n + 1 \drJ
Let us now resume equation (1) of 554, which is relative to the sur
face,
=
2a 2 a
this equation, combined with the preceding ones, gives
p = const. + {(n+a1)r-l} gr(l-m').
At the surface, r is very nearly equal to a ; by making them entirely so,
for the sake of simplicity, we shall have
p = const. + "~ g (1 — m 2)
Let P be the gravity at the equator of the spheroid, and « p
BOOK I.] NEWTON'S PRINC1PJA. 293
the ratio of the centrifugal force to gravity at the equator; we shall
have
p =
whence it follows that, from the equator to the poles, gravity varies as the
square of the sine of the latitude. In the case of nature, where n = — 2,
we have
p = P [I + | ap.m*}',
which agrees with what we have before found.
But it is remarkable that if n =; 3, we have p = P, that is to say, that
if the attraction varies as the cube of the distance, the gravity at the sur
face of homogeneous spheroids is every where the same, whatever may be
the motion of rotation.
581. We have only retained, in the research of the figure of the celestial
bodies, quantities of the order a ; but it is easy, by the preceding analysis,
to extend the approximations to quantities of the order « 2, arid to superior
orders. For that purpose, consider the figure of a homogeneous fluid
mass in equilibrium, covering a spheroid differing but little from a sphere,
and endowed with a rotatory motion ; which is the case of the earth and
planets. The condition of equilibrium at the surface gives, by No. 557,
the equation
const. = V — -§- r * (m 2— £).
i)
The value of V is composed, 1st, of the attraction of the spheroid co
vered by the fluid upon the molecule of the surface, determined by the
coordinates r, 69 and w, 2dly, of the attraction of the fluid mass upon this
molecule. But the sum of these two attractions is the same as the sum of
the attractions, 1st, of a spheroid supposing the density of each of its shells
diminished by the density of the fluid; 2dly, of a spheroid of the same density
as the fluid, and whose exterior surface is the same as that of the fluid.
Let V be the first of these attractions and V" the second, so that
V = V'+V"; we shall have, supposing g of the order a and equal to « g',
const. = V + V" — "-J- . r2 . (m 2 — £).
/it
\Ve have seen in 553 that V may be developed into a series of the form
UW UCD U(2)
r ~ " Tr
U (i) being subject to the equation of partial differences,
0
0=
= - ___
dm 1 — m
T3
294 A COMMENTARY ON [SECT. XII. & XIII.
and by the analysis of 561, we may determine U(i), with all the accuracy
that may be wished for, when the figure of the spheroid is known.
In like manner V" may be developed into a series of the form
U, (i) being subject to the same equation of partial differences as U (i). If
we take for the unit of density that of the fluid, we have, by 561,
U (» — 4* 7 (:) .
U' - (i + 3) (2 i + 1 '
r i + 3 being supposed developed into the series
ZW + ZW + z<2> +&c.
in which Z(i) is subject to the same equation of partial differences, as U(l).
The equation of equilibrium will therefore become
_
i being equal to greater than unity.
If the distance r from the molecule attracted to the center of the sphe
roid were infinite, V would be equal to the sum of the masses of the sphe
roid and fluid divided by r ; calling, therefore, m this mass, we have
U(°) -f U/0) = m. Carrying the approximation only to quantities of the
order a2, we may suppose
r = 1.+ a y + a8 y';
which gives
Suppose
y = Y(1> + Y(-> + Y® + &c.
y/ _ Y'd) + Y'^ + Y'® + &c.
y" = M^ + M«> + M® + &c.
Y &t Yx (i), and M (i) being subject to the same equation of partial differ
ences as U (i) ; we shall have
1 • £
Then observe that U (i) is a quantity of the order a, since it would be
nothing if the spheroid were a sphere ; thus carrying the approximation
only to terms of the order a2, U « will be of this form a U' & + «2 U'' (i).
Substituting therefore these values in the preceding equation of equili
brium, and there changing r into 1 + a y + « 2 y', we shall have to quan
tities of the order « 3,
BOOK I.] NEWTON'S PRINCIPIA. 295
const. = fA [I — ay + «2y2 — «2y'}
"a U' (i) + « 2 U" W — (i + 1) a* y U' «
+ 2
r 2i+ 1
' O /O i
*
'1
_ .
Equating separately to zero the terms of the order «, and those of the
order a2, we shall have the two equations,
, ^J*— \ Y (*) = 2 U' (1) __ -^(m 2 — i) :
2 i + i; 2 ^
Cx being an arbitrary constant. The first of these equations detects Y
and consequently the value of y. Substituting in the second member of
the second equation, we shall develope by the method of No. 560. in a
series of the form
N<°> + NW+ N^ + Sue.
N (i) being subject to the same equation of partial differences as U w, and
we shall determine the constant C' in such a manner that N (0) is nothing;
thus we shall have
N «
Y' & =
4,*
2i + 1
and consequently
The expression of the radius r of the surface of the fluid will thus be
determined to quantities of the order a 3, and we may, by the same process,
carry the approximation as far as we wish. We shall not dwell any longer
upon this object, which has no other difficulty than the length of calcula
tions; but we shall derive from, the preceding analysis this important con
clusion, namely, that we may affirm that the equilibrium is rigorously pos
sible, although we cannot assign the rigorous figure which satisfies it ; for
we may find a series of figures, which, being substituted in the equation of
equilibrium, leave remainders successively smaller and smaller, and which
become less than any given quantity. v
T4
29G A COMMENTARY ON [SECT. XII. & X1I1.
COMPARISON OF THE PRECEDING THEORY WITH OBSERVATIONS.
582. To compare with observations the theory we have above laid down,
we must know the curve of the terrestrial meridians, and those which we
trace by a series of geodesic operations. If through the axis of rotation
of the earth, and through the zenith of a plane at its surface we imagine
a plane to pass produced to the heavens; this plane will trace a great cir
cle which will be the meridian of the plane : all points of the surface of
the earth which have their zenith upon this circumference, will lie under
the same celestial meridian, and they will form, upon this surface, a curve
which will be the corresponding terrestrial meridian.
To determine this curve, represent by u = 0 the equation of the surface
of the earth ; u being a function of three rectangular coordinates x, y, z.
Let x', y', z', be the three coordinates of the vertical which passes through
the place on the earth's surface determined by the coordinates x, y, z ; we
shall have by the theory of curved surfaces, the two following equations,
/d u\ , /du
°=
0=
Adding the first multiplied by the indeterminate >. to the second, we
get
dz' -
—
\dx
This equation is that of any plane parallel to the said vertical : this ver
tical produced to infinity coinciding with the celestial meridian, whilst its
foot is only distant by a finite quantity from the plane of this meridian,
may be deemed parallel to that plane. The differential equation of this
plane may therefore be made to coincide with the preceding one by suita-
blv determining the indeterminate X.
•I
Let
d z' = a d x' + b d /,
be the equation of the plane of the celestial meridian ; comparing it with
the preceding one, we shall get
To get the constants a, b, we shall suppose known the coordinates of
BOOK L] NEWTON'S PRINCIPIA. 297
the foot of the vertical parallel to the axes of rotation of the earth and that
of a given place on its surface. Substituting successively these coordi
nates in the preceding equation, we shall have two equations, by means of
which we shall determine a and b. The preceding equation combined
with that of the surface u = 0, will give the curve of the terrestrial meri
dian which passes through the given plane.
If the earth were any ellipsoid whatever, u would be a rational and
entire function of the second degree in x, y, z ; the equation (a) would
therefore then be that of a plane whose intersection with the surface of the
earth, would form the terrestrial meridian : in the general case, this me
ridian is a curve of double curvature.
In this case the line determined by geodesic measures, is not that of
the terrestrial meridian. To trace this line, we form a first horizontal
triangle of which one of the angles has its summit at the origin of
this curve, and whose two other summits are any visible objects. We de
termine the direction of the first side of the curve, relatively to two sides
of the triangle, and to its length from the point where it meets the side
which joins the two objects. We then form a second horizontal triangle
with these objects, and a third one still farther from the origin of the
curve. This second triangle is not in the plane of the first; it has nothing
in common with the former, but the side formed by the two first objects ;
thus the first side of the curve being produced, lies above the plane of
this second triangle; but we bend it down upon the plane so as always to
form the same angles with the side common to the two triangles, and it is
easy to see that for this purpose it must be bent along a vertical to this
plane. Such is therefore the characteristic property of the curve traced
by geodesic operations. Its first side, of which the direction may be
supposed any whatever, touches the earth's surface; its second side is this
tangent produced and bent vertically ; its third is the tangent of the se
cond side bent vertically, and so on.
If through the point where the two sides meet, we draw in the tangent
plane at the surface of the spheroid, a line perpendicular to one of the
sides, it is clear that it will be perpendicular to the other ; whence it follows?
that the sum of the sides is the shortest line which can be drawn upon the
surface between their extreme points. Thus the lines traced by geodesic
operations, have the property of being the shortest we can draw upon the
surface of the spheroid between any two of their points; andp.294,Vol.I.
they would be described by a body moving uniformly in this surface.
298 A COMMENTARY ON [SECT. XII. & XIII.
Let x, y, z be the rectangular coordinates of any part whatever of the
curve ; x + d x, y + d y, z + d z will be those of points infinitely near to
it. Call d s the element of the curve, and suppose this element produced
by a quantity equal tods; x + 2 d x, y + 2 d y, z + 2 d z will be the
coordinates of extremity of the curve thus produced. By bending it ver
tically, the coordinates of this extremity will become x + 2dx + d2x,
y + 2 d y + d 2 y, z + 2dz-f<i2z; thus — d 2 x, — d2 y, — d °- z
will be the coordinates of the vertical, taken from its foot ; we shall there
fore have by the nature of the vertical, and by supposing that u = 0 is
the equation of the earth's surface,
/d u\ , /d
°= (die)
d u
/ u\ ,
°--= (dx) d
equations which are different from those of the terrestrial meridian. In these
equations d s must be constant; for it is clear that the production of
d s meets the foot of the vertical at an infinitely small quantity of the fourth
order nearly.
Let us see what light is thrown upon the subject of the figure of the earth
by geodesic measures, whether made in the directions of the meridians,' or in
directions perpendicular to the meridians. We may always conceive an ellip
soid touching the terrestrial surface at every point of it, and upon which, the
geodesic measures of the longitudes and latitudes from the point of contact,
for a small extent, would be the same as at the surface itself. If the entire
surface were that of an ellipsoid, the tangent ellipsoid would every where
be the same ; but if, as it is reasonable to suppose, the figure of the meri
dians is not elliptic, then the tangent ellipsoid varies from one country to
another, and can only be determined by geodesic measures, made in diffe
rent directions. It would be very interesting to know the osculating ellip
soids at a great number of places on the earth's surface.
Let u = x " -\- y 2 + z 2 — 1 — 2 a u', be the equation to the surface
of the spheroid, which we shall suppose very little different from a sphere
whose radius is unity, so that a is a very small quantity whose square may
be neglected. We may always consider u' as a function of two variables
x, y ; for by supposing it a function of x, y, z, we may eliminate z by
means of the equation z = V I — x2 — y1. Hence, the three equa
tions found above, relatively to the shortest line upon the earth's surface,
become
BOOK I.]
NEWTON'S PRINCIPIA.
299
d2z — zd2x = a - d J z ;
yd2z — zday = a(d~y)' d'z-
This line we shall call the Geodesic line.
Call r the radius drawn from the center of the earth to its surface, 6 the
angle which this radius makes with the axis of rotation, which we shall
suppose to be that of z, and p the angle which the plane formed by this
axis and by r makes with the plane of x, y ; we shall have
x = r sin. 6. cos. <f> ; y = r sin. 6 sin. p ; z = r cos. 6 ;
whence we derive
r 2 sin. 2 0. dp = xdy — ydx;
— r2 d 6 = (xdz — zdx) cos. p + (y d z — zdy) sin. p
d s2 = dx2+dy2+dz2 = dr2+r2 dd 2+r2d p2 sin. ° 6.
Considering then u', as a function of x, y, and designating by -y the lati
tude ; we may suppose in this function r= 1, and -y= 100°— d, which gives
x = cos. y cos. p ; y = cos. y- sin. p ;
thus we shall have
rd
d u
but we have
-4/
= tan. <p ;
cos.
x 2 _J_ y 2 _ CQS
whence we derive
x d x + y d y ,
d 4/ = = r1- - — r-; d
sin. -y cos. -y
Substituting these values of d -y and of d f in the preceding differential
equation in u', and comparing separately the coefficients of d x and d y ;
we shall have
(d u'\ _ cos. <p /d u\ sin. p
d x / sin. 4/ ' \d -y / cos. 4/
/d u\ sin, g /d u\
\d y / ~ "* sin. 4 ' Vd^J H
which give
xdy — ydx
= - * — ^ -
x2
d u
d~
cos. <p /d u'
cos.
d u\ , „
-T — )d2 y —
d x/ ?
dy
^)d^ = -
v /
sin. -y cos. -y '
,du\
^*«* j — y «
v -l_ ir fl 3 ^'\
^d p / | j t
300 A COMMENTARY ON [SECT. XII. & XIII.
But neglecting quantities of the order «, we have x d 2 y — y d 2 x = 0 ;
and the two equations
xd2z — zd2x = 0, yd2z — zd2y = 0,
give
Z2(xd2x+yd2y)
zd'z= »' + /•
and
x* + y2 + z2 = 1
gives
xd2x + yd2y + zd2z + tls2 = 0;
substituting for z d 2 z its preceding value, we shall have
xd2x + yd2y = — (x2 + y2)ds2 = — d s2cos. 2 ^;
wherefore
d u'\ , /d u'\ , , /d u
The first of equations (O), will thus give by integration,
r2df sin.8* = cds+ ads/ds(|j-^'); ..... (p)
c being the arbitrary constant.
The second of equations (O) gives
d. (x d z — z d x) =r a. (-, — d " z ;
-, —
but it is easy to see by what precedes, that we have
d 2 z = — d s 2. sin. 4/ ;
we have therefore
d (x d z — z d x) = — ads ° (— - -\ sin. 4/ ;
in like manner we have
d (y d z — z d y) = — ads2 f -,-- ^ sin. 4<;
9
we shall therefore have
r " d 0 = c' d s sin. <p + c" d s cos. <p
. C /d u\ /d u\ . , )
— ads cos. <pf d s -( \.Td// °OS* ^ "^" \d — / Sm' ^ ^ (
— ads sin. p/d s| (~)sin. p — (-j^)cos. p tan. -4/j; . (q)
First consider the case in which the first side of the Geodesic line is
parallel to the corresponding plane, of the celestial meridian. In this case
d p is of the order «, as also d r ; we rTave, therefore, neglecting quantities
of the order a2, d s = — r d 0, the arc s being supposed to increase from
BOOK I.] NEWTON'S PRINCIPIA. 301
the equator to the poles. 41 expressing the latitude, it is easy to see that
we have 0 = 100° — 4/ — (TTT) > which gives
d0 = _d^-«d
we have therefore
Thus naming s the difference in latitude of the two extreme points of
the arc s, we shall have
u/ being here the value of u' at the origin of s.
If the earth were a solid of revolution, the geodesic line would be al
ways in the plane of the same meridian ; it departs from it if the parallels
are not circles ; the observations of this deflection may therefore clear up
this important point of the theory of the earth. Resume the equation (p)
and observe that in the present case, d p and the constant c of this equa
tion are of the order a, and that we may there suppose r = 1, d s = d -4/,
6 = 100° — -4/; we shall thus get
d <p cos. 2-\}/ = cd-vJ/ + a
However, if we call V the angle which the plane of the celestial meri
dian makes with that of x, y, whence we compute the origin of the angle
<p; we shall have d x' = tan. V = d y'; x', y', z' being the coordinates
of that meridian whose differential equation, as we have seen in the pre
ceding No., is
d z' = a d x' + b d y'.
Comparing it with the preceding one, we see that a, b are infinite and
Q
such that -- p = tan. V, the equation (a) of the preceding No. thus
gives
/d u\ , /d u
0=(^-).tan.V-(^
whence we derive
We may suppose V = <p, in the terms multiplied by u; moreover
— = tan. <p : w have therefore
x
302 A COMMENTARY ON [SECT. XII. & XIII.
/du\
\T — /
cos. 4> cos. <f> £tan. <p — tan. V] =
cos. 4* cos. <f '
which gives
.
cos. 2 4
The first side of the Geodesic line, being supposed parallel to the plane
of the celestial meridian, the differentials of the angle V, and of the dis
tance (<f> — V) cos. 4- from the origin of the curve to the plane of the
celestial meridian ought to be nothing at this origin ; we have therefore
at this point
a( j — )
\d 0 /
tan.
~^—,
cos. 2 4'
and consequently, the equation (p) gives
u, and 4'/ being referred to the origin of the arc s.
At the extremity of the measured arc, the side of the curve makes with
the plane of the corresponding celestial meridian an angle very nearly
equal to the differential of (p — V) cos. 4/> divided by d 4^ V being sup
posed constant in the differentiation ; by denoting therefore this angle by
», we shall have
d 0 / TT\ •
•a — ~ cos. 4' — (<p — V) sin. 4/-
If we substitute for -~- its value obtained from the equation (p), and for
f — V, its preceding value, we shall have
a f /d u/\ . /d u'\ , , /d u'\ ")
» = -- . < ( T-*-I tan. 4^/ — [~i — ) tan. 4/ + / d 4^ ( -i — ) / ;
cos. <p \\ d <p J Wlp/ rV.d?>/J'
the integral being taken from the origin of the measured arc, to its extre
mity. Call s the difference in latitude of its two extreme points ; £ being
supposed sufficiently small for t z to be rejected, we shall have
a E tan. 4 / /d u\ / d 2 u' \ \
w = -- -Z -! ( — > tan. 4' + ( T - 1— r ) ( 5
cos. 4/ I \d p / r \d p d 4// J
in which the values of 4> TT — ^? and f -, - y— r^must be referred, for the
•*\ dp/' \d f d 4//
greater exactness, to the middle of the measured arc. The angle -a must be
BOOK I.] NEWTON'S PRINCIPIA. 303
supposed positive, when it quits the meridian, in the direction of the in
crements of <p.
To obtain the difference in longitude of the two meridians correspond
ing to the extremities of the arc, we shall observe, that u/, V,, -4/,, and
p/5 being the values of u', V, 4/, and <p, at the first extremity, we have
,— v .= —
d u/x /d u'
d p
/d u\
vd®/
cos. 2 ^ ' ' cos. 2 -4, '
but we have very nearly, neglecting the square of £,
c s /d u/x
c = a -r- tan. ;
cos/ s
1 * •
we shall have, therefore,
V-V = ^-p. ((^tan.
cos. Y (_ » d p /
whence results this very simple equation,
(V — V,) sin. -4>, = »;
thus we may, by observation alone, and independently of the knowledge
of the figure, determine the difference in longitude of the meridians cor
responding to the extremities of the measured arc ; and if the value of the
angle -a is such that we cannot attribute it to errors of observations, we
O
shall be certain that the earth is not a spheroid of revolution.
Let us now consider the case where the first side of the Geodesic line
is perpendicular to the corresponding plane of the celestial meridian. If
we take this plane for that of x, y, the cosine of the angle formed by this
side upon the plane, will be — C X.2 + — ; thus this cosine being no
thing at the origin, we have d x = 0, d z = 0, which gives
d . r sin. 6 cos. <f> = 0 ; d . r cos. 6 = 0 ;
and consequently
r d 6 = r d <p sin. 6 . cos. 6 . tan. <p ;
but we have, to quantities of the order a % d s = r d o sin. 6 ; we shall
have, therefore, at the origin,
d d _ tan. <p . cos. 6
d s r
The constant c", of the equation (q), is equal to the value of x d z —
z d x, at the origin ; it is therefore nothing, and the equation (q) gives at
the origin,
-i— = — r sin. <p ;
d s r 2
304 A COMMENTARY ON [SECT. XII. & XIII.
we have, therefore, observing that p is here of the order a, and that thus
neglecting quantities of the order a 2, we have sin. <p = tan. p,
c' = r, cos. Ofl
the quantities r, and 6/ being relative to the origin ; therefore, if we con
sider that at this origin the angle p is what we have before called it,
/d U/v
\"Ty
p, — V,, and whose value we have found equal to 2 ; we shall
have at this point
The equation (q) then gives
dl
d s2
but we have
dd, /d u'\ sin. -J/.
•3—6 ss a I i — -
d s \ d p / cos. 2 4v '
ives
. cos. 0. d p. /d u/\
/ — f ff I / I •
2 - ry 'ds " '\~d^J'
--j /
d s rx sin. ^
we shall get therefore
^- = (1 - 2 . u/) tan. +/ + . (^) tan. • +,
Observing that at the origin,
— au
-~ - -. — -
d s r7 sin. ^y cos. -
the equation (p) gives
c = r, sin. 6, ;
whence we get
d u/ . d 0. /d u/\
j . 2 a . -=-*- 2 . T— ' cos. 6, a ( -=-i- )
d 2 px d s d s _ ' \ d p/
d s 2 " r7 sin. 6, r, sin. e 0, cos. 2 ^
and consequently
d8 p, /d u/\ 2 — cos. * ^
d s 2 v d p / ' cos. 4 4-,
The equ|tion
gives, by retaining amongst the terms of the order s2, only those which are
independent of a,
II 14 i 2 dJJ/ as / d 2 u/ x
"*'" S'ds "" 2 S ' ds2 " cos. 4/VdpdV'
BOOK I.] NEWTON'S PRINCIPIA. 305
wherefore
1 i « s f /d u/\ / d 2 u
-
The difference ofktitudes at the two extremities of the measured arc,
will therefore give
It is remarkable, that for the same arc, measured in the direction of the
meridian, this function, by what precedes, is equal to — — - ; it may thus
tan. •y/ *
be determined in two .ways, and we shall be able to judge whether the
values thus found of the difference of latitudes, or of the azimuthal
angle *-, are due to the errors of observations, or to the excentricity of the
terrestrial parallels.
Retaining only the first power of s, we have
9 — <?< is not the difference in longitude of the two extremities of the arc
s ; this difference is equal to V — V, ; but we have, by what precedes,
cos.
which gives
— V — (d> -- V) -
os., cos.
wherefore
For greater exactness, we must add to this value of V _ V7 the term
depending on s 3, and independent of a, which we obtain in the hypothesis
of the earth being a sphere. This term is^ equal to — A s3.
thus we have
tan> '
cos.
It remains to determine the azimuthal angle at the extremity of the
arc s. For that purpose, call x', and y', the coordinates x, y, referred to
VOL. 1J.
306 A COMMENTARY ON [SECT. XII. & XIII.
the meridian of the last extremity of the arc s ; it is easy to see that the
V d x' 2 + d z 2
cosine of the azimuthal anjjle is equal to \ . If we refer
d s
the coordinates x, y, to the plane of the meridian corresponding to the
first extremity of the arc ; its first side being supposed perpendicular to
the plane of this meridian, we shall have
*2i o-i^' n.l£
d s • ' d s - ' d s •
wherefore, retaining only the first power of s,
d_x d2 x, dj5 d^_z,
d7 : * Ts^ dl - s' dTe5»
but wre have
x' = x cos. (V — V,) + y sin. (V — V,) ;
thus V — V, being, by what precedes, of the order a, we shall have
"dT = S'~d7^+ (V~~ V')~dT*
Again, we have
x = r sin. 6 cos. p ; z = r cos. & ;
we therefore shall obtain, rejecting quantities of the order a 2, and observ
ing that p., -T — /, and j— ' are quantities of the order «,
d s d s
d2x. d2u/ . d20. d <?*
—, — f = a . —, — % sin. 6 + r . -j — { cos. 6 — r sin. ^ . - .- '- .
ds2 ds2 'ds2 ds8
Thence we have
d'u/ /d2u/\dp,8
-— ~ — ^
)up/ /uu/x u »/
d sz Wl-vp/ds2 cos.
moreover, d s = r, sin. 6/ . d tp , ; we shall, therefore, have by substitutijig
dp. , d 2 d, , . ,.
for r,, 0,, -r1' , and -, — „-, their preceding values,
d s d s 2 '
d 2 x ., sin. ~ -^f>. /d u/\
i — zz ( 1 — a u/) • • r~ + a ( j — r ) tan. z -J/, sin. Jc.
d s z cos. ^ \d -y /
r • 1 1 — a u. + a \-\ — -f- ) tan. -vL. f +
cos. v/ *- Ml Y / cos.
Neglecting the superior powers of s, we have, as we have seen,
V V — ? J /d u
T — ~ » . "t»\l /. /*-*U/ti 1
COS. Y/ f • — a l1/ "T a\T — r" J ^an- "r. —
and -— - = 1 ; we therefore have
BOOK I.] NEWTON'S PRINCIPIA. 307
x/ /i A8'0-2^/, /"du/v . /d 2 u/x sin. * 4.
— ' - = s(l— au/) r^-f-as( , / )tan.24..sin. 4,— «s(-, — M r-f/j
s x cos. 4y vdJ*/ v d p2/ cos. 34,'
dx, ,. A sin. 24 /du
-,—
ds
in like manner we shall find
dz . .
the cosine of the azimuthal angle, at the extremity of the arc s, will thus
be
s tan.
This cosine being very small, it may be taken for the complement of
the azimuthal angle, which consequently is equal to
100° — s tan
( «rd'uAi
. 4X , , , /du/x . \dp*} V.
V -a<+H^)tan^--^sTM7/
C /d 2 u/x -\
•w , , , /du/Nf , aVdp2; v.
T/ J 1 — a u/+a ( -j-^- ) tan. ^ i!^1 — (
I. \d 4 / cos.2 4-, )
For the greater exactness, we must add to this angle that part depend
ing on s3, and independent of «, which we obtain in the hypothesis of the
earth's sphericity. This part is equal to £ s 3 (| + tan. z 41/) tan. 41/, Thus
the azimuthal angle at the extremity of the arc s is equal to
100°-stan.4
The radius of curvature of the Geodesic line, forming any angle what
ever with the plane of the meridian, is equal to
ds2
V (d2x)2 + (d2y)2+ (d*z)2'
d s being supposed constant; let R be this radius. The equation
x2 + y2 + z2=l + 2«u/ gives
xd2x+ yd2y + zd2z = — d s2 + ad8u';
if we add the square of this equation to the squares of equations (O), we
shall have, rejecting terms of the order a2,
(x«+ y2 + z2) £(d2x)2+ (d2y)2+ (d2z)2}=ds4 — 2ads2d2u'
whence we derive
d2 u'
R = 1 + au' + a-^y.
In the direction of the meridian, we have
d!u'
wherefore
U2
308 A COMMENTARY ON [SECT. XII. & XIII.
In the direction perpendicular to the meridian, we have by what pre
cedes,
wherefore
rd2u
R =
d ,
c
If in the preceding expression of V — V, , we make -^5- = s', it takes
this very simple form relative to a sphere of the radius R,
V — V, = — ^r-. -f 1 l-sfz. tan.8 >}>,{•.
cos. 4V' i. • ™ J
The expression of the azimuthal angle becomes
100° — s' tan. -4>, [I — i s/2 (J + tan.2 •^/}}.
Call X, the angle which the first side of the Geodesic line forms with the
plane corresponding to the celestial meridian, we shall have
u /u p _
2 = Vd^J dT2+ Vdf^/ dV+ Wl p2^ ds2-t \dpdV ds d s^ Vd^V d s2'
But supposing the earth a sphere, we have
dft _ sin. X . d.'p, _ 2 sin. X cos. X ^ ^ .
d s ~ cos. ^ ' d s 2 " cos ^
wherefore,
5'- = cos. X ; = — sin. 2 X tan.
d s d s2
_ sn...tan
"
_
ds2" cos.
the radius of curvature R, in the direction of this Geodesic line, is there
fore
To abridge this, let
d 2 U
K =
r
BOOK I.] NEWTON'S PRINCIPIA. 309
A =
R = K + A sin. 2 X + B cos. 2 X.
The observations of azimuthal angles, and of the difference of the lati
tudes at the extremities of the two geodesic lines, one measured in the
direction of the meridian, and the other in the direction perpendicular to
the meridian, will give, by what precedes, the values of A, B and K ; for
the observations give the radii of curvature in these two directions. Let
R, and R' be these radii ; we shall have
R' + R"
~2~
R R' — R"
B= ~~2 - ;
and the value of A will be determined, either by the azimuth of the ex
tremity of the arc measured in the direction of the meridian, or by the
difference in latitude of the two extremities of the arc measured in a di
rection perpendicular to the meridian. We shall thus get the radius of
curvature of the geodesic line, whose first side forms any angle whatever
with the meridian.
j±
If we call 2 E, an angle whose tangent is-^-, we shall have
R = K + VA" + B2. cos. (2 X — - 2 E) ;
the greatest radius of curvature corresponds with X =r E ; the correspond
ing geodesic line forms therefore the angle E, with the plane of the me
ridian. The least radius of curvature corresponds with X = 100°+ E;
let r be the least radius, and r' the greatest, we shall have
R = r + (r — r) cos. 2 (X — E),
X — E being the angle which the geodesic line corresponding to R, forms
with that which corresponds with r'.
We have already observed, that at each point of the earth's surface,
we may conceive an osculatory ellipsoid upon which the degrees, in all
directions, are sensibly the same to a small extent around the point of os
culation. Express the radius of this ellipsoid by the function
1 — a sin. 2 %J/ Jl + h cos. 2 (<p + /3)j,
the longitudes <p being reckoned from a given meridian. The expression
us
310 A COMMENTARY ON [SECT. XII. & XIII.
of the terrestrial meridian measured in the direction of the meridian,
will be, by what precedes,
£ — ^ . [I + h cos. 2 (<p + /5)J . { 1 + 3 cos. 2 -4/ — 3 * sin. 2 4}.
If the measured arc is considerable, and if we have observed, as in
France, the latitudes of some points intermediate between the extremity;
we shall have by these measures, both the length of the radius taken for
unity, and the value of « {1 + h cos. 2 (p + 13)}. We then have, by
what precedes,
, tan. 8 4 (1 + cos. 2 40 . .
» = — 2 « h . E . — — i *—. — . sin. 2 (a + £) ;
cos. 4
the observation of the azimuthal angles at the two extremities of the arc
will give a h sin. 2 (<p + /3). Finally, the degree measured in the direc
tion perpendicular to the meridian, is
1° + 1°. ajl + h cos. 2 (<f> + 13)} sin.2 4 + 4°. ah tan.2 $ cos. 2 (<p + £);
the measure of this degree will therefore give the value of « h sin. 2 (p + ,6).
Thus the osculatory ellipsoid will be determined by these several mea
sures : it would be necessary for an arc so great, to retain the square of e
in the expression of the angle »; and the more so, if, as it has been ob
served in France, the azimuthal angle does not vary proportionally to
the measured arc: at the same time we must add a term of the form
« k sin. 4- cos. 4/ sin. (<p -f /3'), to get the most general expression of this
radius.
583. The elliptic figure is the most simple after that of the sphere : we
have seen above that this ought to be the figure of the earth and planets,
on the supposition of their being originally fluid, if besides they have
retained their primitive figure. It was natural therefore to compare
with this figure the measured degrees of the meridian; but this compari
son has given for the figure of the meridians different ellipses, and which
disagree too much with observations to be admissible. However, before we
renounce entirely the elliptic, we must determine that in which the greatest
defect of the measured degrees, is smaller than in every other elliptic
figure, and see whether it be within the limits of the errors of observations.
O 7
We arrive at this by the following method.
Let a (1), a(2) , a (3), &c. be the measured degrees of the meridians ; p (1),
p (2\ p C3), &c. the squares of the sines of the corresponding latitudes :
suppose that in the ellipse required, the degree of the meridian is expressed
by the formula z + p y ; calling x (1), x (2), x (3>, &c. the errors of observation,
we shall have the following equations, in which we shall suppose that p (1)s
p v% p ®, &c. form an increasing pi-ogression,
BOOK I.] NEWTON'S PRINCIPIA. 311
a") — z — pOy = x«)
a (2) — z — p«>y = x<2> ...... (A)
a(n) z p(n) y __ x(n)
n being the number of measured degrees.
We shall eliminate from these equations the unknown quantities z and y,
and we shall have n — 2' equations of condition, between the n errors
x (1), x (2-, x (n). We must, however, determine that system of errors,
in which the greatest, abstraction being made of the signs, is less than in
every other system.
First suppose that we have only one equation of condition, which may
be represented by
a = m x (1) + n x {2> + p x f3) + &c.
a being positive. We shall have the system of the values of x (1), x (2), &c.
which gives, not regarding signs, the least value to the greatest of them ;
supposing them all nearly equal, and to the quotient of a divided by the
sum of the coefficients, m, n, p, &c. taken positively. As to the sign
which each quantity ought to have, it must be the same as that of its co
efficient in the proposed equation.
If we have two equations of condition between the errors, the system
which will give the smallest value possible to the greatest of them will be
such that, signs being abstracted, all the errors will be equal to one ano
ther, with the exception of one only which will be smaller than the rest,
or at least not greater. Supposing therefore that x (1) is this error, we
shall determine it in function x (2), x (3), &c. by means of one of the proposed
equations of condition ; then substituting this value of x (1) in the other
equation of condition, we shall form one between x {2), x (3), &c. ; which re
present by the following
a = m x ® + n x (3> + &c.
a being positive; we shall have, as above, the values of x(2), x(3), &c. by
dividing a by the sum of the coefficients m, n, &c. taken positively, and by
giving successively to the quotient the signs ofm, n, &c. These values sub
stituted in the expression of x (1) in terms of x (% x (3), &c. will give the value
of x U) ; and if this value, abstracting signs, is not greater than that of x (2),
this system of values will be that which we must adopt; but if greater, then
the supposition that x (1) is the least error, is not legitimate, and we must
successively make the same supposition as to x (2>, x (3), &c. until we arrive
at an error which is in this respect satisfactory.
If we have three equations of condition between the errors ; the system
which will give the smallest value possible to the greatest of them, will be
U4
312 A COMMENTARY ON [SECT. XII. & XIII.
such, that, abstracting signs, all the errors will be equal, with exception of
two, which will be less than the others.
Supposing therefore that x (1), x ('2) are these two errors, we shall elimi
nate them from the third of the equations of condition by means of the
other two, and we shall have an equation between the errors x (3), x(1), &c.:
represent it by
a = m x ® + n x W> +' &c.
a being positive. We shall have the values of x (3), x (1), &c. by dividing
a by the sum of the coefficients m, n, &c. taken positively, and by giving
successively to the quotient, the signs of m, n, &c. These values substi
tuted in the expressions of x (1), and of x (2) in terms of x ®, x W, &c. will
give the values of x (1), and of x (2), and if these last values, abstracting
signs, do not surpass x (3\ we shall have the system of errors, which we
ought to adopt; but if one of these values exceed x(3), the supposition that
x (1), and x ^ are the smallest errors is not legitimate, and we must use
the same supposition upon another combination of errors x(1), x(2), £c.
taken two and two, until we arrive at a combination in which this suppo
sition is legitimate. It is easy to extend this method to the case where
we should have four or more equations of condition, between the errors x (l\
x (2), &c. These errors being thus known, it will be easy to obtain the
values of z and y.
The method just exposed, applies to all questions of the same nature ;
thus, having the number n of observations upon a comet, we may by this
means determine that parabolic orbit, in which the greatest error is, ab
stracting signs, less than in any other parabolic orbit, and thence recog
nise whether the parabolic hypothesis can represent these observations.
But when the number of observations is considerable, this method be
comes too tedious, and we may in the present problem, easily arrive at
the required system of errors, by the following method.
Conceive that x (i), abstracting signs, is the greatest of the errors
x (1), x ®, &c. ; we shall first observe, that therein must exist another error
x (l\ equal, and having a contrary sign to x (i) ; otherwise we might, by
making z to vary properly in the equation
a (i) — z — p (i). y = x w,
diminish the error x w, retaining to it the property of being the extreme
error, which is against the hypothesis. Next we shall observe that x w
and x (i/) being the two extreme errors, one positive, and the others nega
tive, and equal to one another, there ought to exist a third error x (l">,
equal, abstracting signs, to x (i). In fact, if we take the equation corre-
BOOK I.] NEWTON'S PRINCIPIA. 313
spending to x (i', from the equation corresponding to x (i/), we shall
have
a 00 _ a W — {p V _ p WJ. y - x M — x «.
The second member of this equation is, abstracting signs, the sum of
the extreme errors, and it is clear, that in varying y suitably, we may di
minish it, preserving to it the property of being the greatest of the sums
which we can obtain by adding or subtracting the errors x(1), x(2), &c.
taken two and two ; provided there is no third error x (i"> equal, abstract
ing signs, to x (i> ; but the sum of the extreme errors being diminished,
and these errors being made equal, by means of the value of z, each of
these errors will be diminished, which is contrary to the hypothesis.
There exists therefore three errors x (i), x (i/), x (i//) equal to one another,
abstracting signs, arid of different signs the one from the other two.
Suppose that this one is x ^ ; then the number i' will fall between the
two numbers i and i". To show this, let us imagine that it is not the
case, and that i' is below or above both the numbers i, i". Taking the
equation corresponding to V, successively from the two equations corre-^
spending to i and to i", we shall have
a 0) _ a M — (p « _ p(i')) y - x W _ XM;
a G") _ a GO — (p (i"> — p (i/)) y = x ^ — x «.
The second members are equal and have the same sign ; these are also,
abstracting signs, the sum of the extreme errors; but it is evident, that
varying y suitably, we may diminish each of these sums, since the coeffi
cient of y, has the same sign in the two first members : moreover, we may,
by varying z properly, preserve to x(i/) the same value; x w and x(i';) will
therefore then be, abstracting signs, less than x (i/) which will become the
greatest of the errors without having an equal ; and in this case, we may,
as we have seen, diminish the extreme error ; which is contrary to the hy
pothesis. Thus the number i7 ought to fall between i and i".
Let us now determine which of the errors x (1), x (2), &c. are the extreme
errors. For that purpose, take the first of the equations (A) successively
from the following ones, and we shall have this series of equations,
aw _ a w — (p<2) — p(1)) y = x(2> — x<»,
aC3)__a(1> — (p(3> — p«>) y = x13' — x«; . . . . (13)
&c.
Suppose y infinite ; the first members of these equations will be nega
tive, and then the value of x(n will be greater than x(2), x(3), &c. : dimin
ishing y continually, we shall at length arrive at a value that will render
positive one of the first members, which, before arriving at this state, will
314 A COMMENTARY ON [SECT. XII. & XIII.
be nothing. To know which of these members first becomes equal to zero,
we shall form the quantities,
a(2)_aU) a(3)_a(D a(4)_a(D
p u; " p w _ p u; ' p w P
o (r) _ o '
_p(D
Call ]3 (1^ the greatest of these quantities, and suppose it to be
if there are many values equal to /3W, we shall consider that which cor
responds to the number r the greatest, substituting (3(li for y, in the
(r — l)th of the equations (B), x (r) will be equal to x(1\ and diminishing
y, it will be equal to x (1), the first member of this equation then becoming
positive. By the successive diminutions of y, this member will increase
more rapidly than the first members of the equations which precede it ;
thus, since it becomes nothing when the preceding ones are still nega
tive, it is clear that, in the successive diminutions of y, it will always be
the greatest which proves that x(r> will be constantly greater than x(1),
x(2), . . . x^-1), when y is less than /S(1).
The first members of the equations (B) which follow the (r — l)th will
be at first negative, and whilst that is the case, x(r + 1), x(r + 2), &c. will be
less than x(1), and consequently less than x(r), which becomes the greatest
of all the errors x(I), x(2), .. . x("), when y'begins to be less than /3(1). But
continuing to diminish y, we shall get a value of it, such that some of the
errors x (r + J), x (r + 2), &c. begin to exceed x (r).
To determine this value of y, we shall take the rth of equations (A) suc
cessively from the following ones, and we shall have
a(r + D _ a(r) _ Jp(r + l) _ p (r)J y _ x (r + 1) __ x (r) .
a(r + 2) a(r) Jp(r + 2) p WJ y — x (r + 2) x (r)^
Then we shall form the quantities
a (r + 1) a (r) a (r + 2) a (r)
• • 5irp
p(r+l) pW p(r + 2) p(r)'
Call /3®, the greatest of these quantities, and suppose that it is
." (r/ ~ — £. : if many of these quantities are equal to (3 (2), we shall suppose
that r' is the greatest of the numbers to which they correspond. Then x W
will be the greatest of the errors x (1), x (2), &c. . . . x (n) so long as y is com
prised between /3 (1), and /3 (2) ; but when by diminishing y, we shall arrive at
6(2) ; then x (r/) will begin to exceed x w, and to become the greatest of the
errors.
To determine within what limits we shall form the quantities
Let /3W> be the greatest of these quantities, and suppose that it is
BOOK I.] NEWTON'S PRINCIPIA. 315
^ _
(1 . j* - — <yj : if several of the quantities are equal to /3 (3\ we shall sup
pose that r" is the greatest of the numbers to which they correspond, x (r)
will be the greatest of all the errors from y = /3^, to y = /3(3>. When
y = /3(3), then x(r") begins to be this greatest error. Thus preceding, we
shall form the two series,
oo; j8W; J3«; /SC3) ; . . . /3«D ;_a> ; ..... (C)
The first indicates the errors x(1), x(r\ x1^, &c. which become succes
sively the greatest : the second series formed of decreasing quantities, in
dicates the limits of y, between which these errors are the greatest; thus,
x^ is the greatest error from y = cc, to y = j8W ; x W is the greatest er
ror from y = (3(l\ to y = /3(2); x^ is the greatest error from y = /3^,
to y = /3(3), and so on.
Resume now the equations (B) and suppose y negative and infinite.
The first members of these equations will be positive, x ^ will therefore then
be the least of the errors x^, x(2), &c. : augmenting y continually, some
of these members will become negative, and then x(1) will cease to be the
least of the errors. If we apply here the reasoning just used in the case
of the greatest errors, we shall see that if we call xW the least of the
quantities
a(s) a(l)
and if we suppose that it is — —} ^ , s being the greatest of the num
bers to which X(l) corresponds, if several of these quantities are equal to
XOj x(i) -will be the least of the errors from y = — oc, to y = X('\ In
like manner if we call X(2' the least of the quantities
a(s + !) — a^ a(s+2) — a(s) 0
(, + 0 — (5)5 (8 + 2) — ft', &c.
O (** / _^ *1 (*V
and suppose it to be — T^ ^j , s' being the greatest of the numbers to
which X<2) corresponds, if several of these quantities are equal to xW; x^
will be the smallest of the errors from y = X^, to y r= X^; and so forth.
In this manner we shall form the two series
x jx ;x ;x -';...x^p''
— x; XO; xW; XW;...X^; oo ; (D)
The first indicates the errors xW, x<">, x^8'>, &c. which are successively
the least as we augment y : the second series formed of increasing terms,
indicates the limits of the values of y between which each of these errors
316 A COMMENTARY ON [SECT. XII. & XIII.
is the least; thus xri> is the least of the errors fromy = — GO, to y = X^
x(s) is the least of the errors, from y = X^, to y = X(% and thus of the
rest.
Hence the value of y which, to the required ellipse, will be one of the
quantities /3W, jSW, j3^; &c. X^, X<2>, &c. ; it will be in the first series,
if the two extreme errors of the same sign are positive. In fact, these
two errors being then the greatest, they are in the series x^, xw, xw,
&c. ; and since one and the same value of y renders them equal they
ought to be consecutive, and the value of y which suits them, can only
be one of the quantities /3^, /S^, &c. ; because two of these errors cannot
at the same tune be made equal and the greatest, except by one only of
these quantities. Here, however, is a method of determining that of the
quantities $^\ /3(2), &c. which ought to be taken for y.
Conceive, for example, that /3(3) is this value; then there ought to be
found by what precedes between x^, and x(l% an error which will be the
minimum of all the errors, since x(r/), and x(l"; will be the maxima of these
errors; thus in the series x^, x^, x^'-*, &c. soma one of the numbers
s, s', &c. will be comprised between r and r'. Suppose it to be s. That
x (s) may be the last of the value of y, it ought to be comprised between
X (1) and X (2> ; therefore if |3 ® is comprised by these limits, it will be the
value sought of y, and it will be useless to seek others. In fact, suppose
we take that of the equations (A), which answers to x (s) successively from
the two equations which respond to xtr/) and to x(r"}; we shall have
ado _ aW — {p« — p M} y = x<r'> — x« ;
a (r") _ a (s) _ lp (r") _ p (s)^y =- x (r") _ x (s).
All the members of these equations being positive, by supposing
y = ft (3), it is clear, that if we augment y, the quantity x (r/) — x (s) will
increase ; the sum of the extreme errors, taken positively, will be there
fore augmented. If we diminish y, the quantity x (r") — x (s) will be aug
mented, and consequently also the sum of their extremes ; /3 (3> is therefore
the value of y, which gives the least of these sums; whence it follows that
it is the only one which satisfies the problem.
We shall try in this way the values of /3 (1), j3 (2), (3 W, &c., which is easily
done by inspection ; and if we arrive at a value which fulfils the preced
ing conditions, we shall be assured of the value required of y.
If any of these values of j8 does not fulfil these conditions, then this
value of y will be some one of the terms of the series x^, XC2), &c. Con
ceive, for example, that it is X ®, the two extreme errors x Cs) and x ^ will
then be negative, and it will have, by what precedes, an intermediate error,
BOOK I.] NEWTON'S PRINCIPIA. 317
which will be a maximum, and which will fall consequently in the series
x (1), x (r\ x (r/), &c. Suppose that this is x to, r being then necessarily
comprised between s and s'j X(2) ought, therefore, to be comprised be
tween j8 (I> and /S (2). If that is the case, this will be a proof that X ® is the
value required of y. We shall try thus all the terms of the series X<2), x®,
X W, &c. up to that which fulfils the preceding conditions.
When we shall have thus determined the value of y, we shall easily ob
tain that of z. For this, suppose that J3 (2> is the value of y, and that the
three extreme errors are x (r), x (r/), x (s) ; we shall have x (s) = — x to? and
consequently
a (r) z p (r). y — X to •
a to — z — p (s). y = — x (r) ;
whence we get
to + a (s) p (r) + p (s)
rj _ __
Z_
2 2
then we shall have the greatest error x W5 by means of the equation
a W _ a to p (s) _ p (r)
X to = _1_ £. E. v.
2 2 y
584. The ellipse determined in the preceding No. serves to recognise
whether the hypothesis of an elliptic figure is in the limits of the errors of
observations ; but it is not that which the measured degrees indicate witli
the greatest probability. This last ellipse, it seems, should fulfil the
following conditions, viz. 1st, that the sum of the errors committed in the
measures of the entire measured arcs be nothing : 2dly, that the sum of
these errors, all taken positively, may be a minimum. Thus considering
the entire ones instead of the degrees which have thence been deduced,
we give to each of the degrees by so much the more influence upon the
ellipticity which thence results for the earth, as the corresponding arc is
considerable, as it ought to be. The following is a very simple method
of determining the ellipse which satisfies these two conditions.
Resume the equations (A) of 589, and multiply them respectively
by the numbers which express how many degrees the measured arcs
contain, and which we will denote by i (l\ i (2), i (3), &c. Let A be the sum
of the quantities i (1). a (1), i (2). a (2), &c. divided by the sum of the numbers
i (1), i (2), &c. ; let, in like manner, P denote the sum of the quantities
i (1). p (1>, i (2l p (2), &c. divided by the sum of the numbers i W, i (2), &c. ;
the condition that the sum of the errors i (1). x (l), i (2). x (-\ &c. is nothing,
gives
0 = A — z— P.y.
318 A COMMENTARY ON [SECT. XII. £ XIII.
If we take this equation from each of the equations A of the preceding
No., we shall have equations of the following form : i
b (1> b ®
Form the series of quotients — ^ , —^ , &c. and dispose them according
to their order of magnitude, beginning with the greatest ; then multiply
the equations O, to which they respond, by the corresponding numbers
i (1), i (2), &c. ; finally, dispose these thus multiplied in the same order as
the quotients.
The first members of the equations disposed in this way, will form a
series of terms of the form
hWy — c«; h^y — c®; h®y — c^;&c. . . . (P)
in which we shall suppose h (1), h ^ positive, by changing the sign of the
terms when y has a negative coefficient. These terms are the errors of
the measured arcs, taken positively or negatively.
Then it is evident, that in making y infinite, each term of this series
becomes infinite ; but they decrease as we diminish y, and end by being
negative — at first, the first, then the second, and so on. Diminishing y
continually, the terms once become negative continue to be so, and de
crease without ceasing. To get the value y, which renders the sum of
these terms all taken positively a minimum, we shall add the quantities
h (1), h (2), &c. as far as when their sum begins to surpass the semi-sum of
all these quantities ; thus calling F this sum, we shall determine r such
that
+ h<2> + h^ + ____ + h « > 3 F;
---- + h^-') < F.
C(r)
We shall then have y = r-^-, , so that the error will be nothing rela
tively to the same degree which corresponds to that of the equations (O),
of which the first member equated to zero, gives this value of y.
To show this, suppose that we augment y by the quantity 3 y, so that
c W c (r - ') c fr)
r— } + 3 y may be comprised between • (r — |j and j— -}. The (r — 1) first
c 0>
terms of the series (P) will be negative, as in the case of y = rrrj; hut in
taking them with the sign +, their sum will decrease by the quantity
jhd) + hOO ____ h ('-')} 3y.
BOOK I.] NEWTON'S PRINCIPIA. 319
c(>-)
The first term of this series, which is nothing when y = T-T- , will be
come positive and equal to h ^ d y ; the sum of this term and the follow
ing, which are positive, will increase by the quantity
{hW+ h<r + « + &c.} ay;
but by supposition we have
hO) + hw ____ h'r- '> < h« + h (' + " + &c. ;
the entire sum of the terms of the series (P), all taken positively, will
therefore be augmented, and as it is equal to the sum of the errors
i(i). x(|) + i&. x(2), &c. of the entire measured arcs, all taken with the
c(»)
sign + , this last sum will be augmented by the supposition of y = r-^} + & y.
It is easy to prove, in the same way, that by augmenting y, so as to be
c(r-l) c(r-2j c(r-2) c (r - 3)
comprised between ,— - n and T- - — -. , or between 7-7- — ^ and ,-7- — ~ , &c.
fi (r — i/ fi (f — */* [\(r — ~) [i\.r — 6)
the sum of the errors taken with the sign + will be greater than when
c«
= HW-
c(r)
Now diminish y by the quantity 5 y so that r-^j — 5 y may be comprised
c(r) C^ + 1)
between 1-7- and T-T- — rr, the sum of the negative terms of the series (P)
n w h ^ t */
will increase, in changing their sign, by the quantity
{h''1) + h(2> + ____ h<r>} 3y;
and the sum of the positive terms of the same series will decrease by the
quantity
Jh(r + 1) + h<r + 2) + &c.} ay;
and since we have
h«) + hW + ____ h« > h^r+J) + h('+2) + &c.,
it is clear that the entire sum of the errors, taken with the sign +, will be
augmented. In the same manner we shall see that, by diminishing y, sp
that it should be between . (r 1} and r-(f-^T) » or Between . (r+it; and . ^,.
&c. the sum of the errors taken with the sign + is greater than when
c(0
y = p— -; this value of y is therefore that which renders this sum a
minimum.
320
A COMMENTARY ON [SECT. XII. & XIII.
The value of y gives that of z by means of the equation
z = A — P . y.
The preceding analysis being founded on the variation of the degrees
from the equator to the poles, being proportional to the square of the sine
of the latitude, and this law of variation subsisting equally for gravity, it
is clear that it applies also to observations upon the length of the seconds'
pendulum.
The practical application of the preceding theory will fully establish its
soundness and utility. For this purpose, ample scope is afforded by the
actual admeasurements of arcs on the earth's surface, which have been
made at different times and in different countries. Tabulated below you
have such results as are most to be depended on for care in the observa
tions, and for accuracy in the calculations.
Latitudes.
Lengths of Degrees.
Where made.
By whom made.
o°.oooo
37 .0093
43 .5556
47 .7963
51 .3327
53 .0926
73 .7037
25538R.85
25666.65
25599.60 '
25640 .55
25658 .28
25683 .30
25832 .25
Peru.
Cape of Good Hope
Pennsylvania.
Italy.
France.
Austria.
Laponia.
Bouguer.
La Caille.
Mason & Dixon.
Boscovich & le Maire.
Delambre & Mechain.
Liesganig.
Clairaut, &c.
SUPPLEMENT
TO
BOOK III.
FIGURE OF THE EARTH.
585. IF a fluid body had no motion about its axis, and all its parts were
at rest, it would put on the form of a sphere ; for the pressures on all the
columns of fluid upon the central particle would not be equal unless they
were of the same length. If the earth be supposed to be a fluid body,
and to revolve round its axis, each particle, besides its gravity, will be
urged by a centrifugal force, by which it will have a tendency to recede
from the axis. On this account, Sir Isaac Newton concluded that the
earth must put on a spheroidical form, the polar diameter being the
shortest. Let P E Q represent a section of the earth, P p the axis, E Q
the equator, .(b m) the centrifugal force of a part revolving at (b). This
force is resolved into (b n), (n m), of which (b n) draws fluid from (b)
to Q, and therefore tends to dim 
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